The Dirac equation in an external magnetic field in the context

Transcription

Lorenzo Menculini
The Dirac equation in an external
magnetic field in the context of
minimal length theories
Tesi di laurea triennale
Universitá degli Studi di Perugia
Settembre 2012
Università degli Studi di Perugia
Facoltà di Scienze Matematiche, Fisiche e Naturali
Corso di laurea triennale in Fisica
Tesi di laurea triennale in Fisica
The Dirac equation in an external magnetic field in
the context of minimal length theories
Relatore
Dott. Orlando Panella
Laureando
Lorenzo Menculini
Correlatore
Prof. Pinaki Roy
ANNO ACCADEMICO 2011/2012
Sessione di Laurea 27 settembre 2012
Alla mia famiglia,
e a John, Reddie e Don Francesco.
iv
...he who has reached the stage where
he no longer wonders about anything, merely demonstrates
that he has lost the art of reflective reasoning.
[Max Planck]
vi
Sommario
Nel presente lavoro viene affrontato lo studio dell’equazione di Dirac in due dimensioni, in
presenza di un campo magnetico statico ed uniforme, all’interno dello scenario del principio di
indeterminazione generalizzato (GUP) con lunghezza minima. Si mostra che con la scelta del
gauge simmetrico per il potenziale vettore e utlizzando coordinate polari nello spazio dei momenti
il problema puó essere risolto esattamente. Si analizzano le differenze rispetto al problema nella
meccanica quantistica ordinaria.
Abstract
In the present work we study the two-dimensional Dirac equation under a static uniform magnetic
field within the scenario of the generalized uncertainty principle (GUP) with minimal length.
We show that by the choice of the symmetric gauge for the vector potential and the use of polar
coordinates in momentum space the problem can be exactly solved. The differences arising with
respect to the ordinary-QM problem are analysed.
Contents
I
Introduction
1
The background
4
1 The Dirac equation
5
1.1
Beyond the Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Dirac’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.1
Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3.2
Probability current density . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.3
Covariant formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2 Generalized uncertainty principle
2.1
2.2
II
14
Heisenberg’s uncertainty relation . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.1
Uncertainty in ordinary quantum mechanics . . . . . . . . . . . . . . . . .
14
2.1.2
Position and momentum eigenstates . . . . . . . . . . . . . . . . . . . . .
15
Minimal length models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.1
A modified inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.2
Minimal length algebra in one dimension . . . . . . . . . . . . . . . . . .
18
2.2.3
Gaining information on position. Maximally localized states. . . . . . . .
21
2.2.4
Extension to higher dimensional spaces . . . . . . . . . . . . . . . . . . .
27
Solving the problem
30
3 Relativistic electrons in a static uniform magnetic field
3.1
31
Setting up the (2+1) Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.1.1
Choice of the gauge. Polar coordinates. . . . . . . . . . . . . . . . . . . .
32
3.2
Solution within ordinary quantum mechanics . . . . . . . . . . . . . . . . . . . .
34
3.3
Solution within a minimal length GUP . . . . . . . . . . . . . . . . . . . . . . . .
36
3.4
Massless case and application to graphene . . . . . . . . . . . . . . . . . . . . . .
47
Conclusions
49
A Expressions in polar coordinates
50
viii
B Angular momenta in a (2+1)-dimensional Dirac theory
52
C Hypergeometric functions
55
Bibliography
58
Acknowledgements
60
Ringraziamenti
62
ix
Notation
Throughout this work we adopt the Gaussian unit system.
c
2.998 × 1010
h
6.626 × 10−27
cm
s
erg · s
~
1.055 × 10−27
erg · s
Dirac’s constant
e
4.803 × 10−10
esu
electron charge
M
9.109 × 10−28
electron mass
G
6.674 × 10−8
g
cm3
g · s2
speed of light
Planck’s constant
Newton’s gravitational constant
x
Introduction
One of the greatest challenges in Physics nowadays is to conciliate quantum mechanics with
general relativity. In fact, while the latter is a geometric theory which treats spacetime as a
continuum (mathematically speaking, a manifold), the first shall lead to a quantum theory of
gravitation which still has to find a coherent formulation, because quantizing the gravitational
field arises serious normalizability problems which constitute a major hindrance to the development of the subject. In other words, on one hand we have to find a way to quantize gravity if
our claim and belief that quantum mechanics correctly describes nature is to be carried on; on
the other, the striking success of Einstein’s theory in accounting for many phenomena means
whe shall take it as a firm point, though in need to be integrated and extended to meet the
requirements of the first approach: finding the way in which this could be achieved is at present
an object of great efforts and study, because merging the two points of view into a unique framework appears to be an highly nontrivial task.
A possible step in the direction of fixing the incompatibility between these two theories has
been hypothesized to be the introduction of a nonzero minimal observable length appearing
when treating systems whose characteristic quantities have a magnitude of the order those of
the Planck scale. This scale bears the name of him who in 1899 first noticed that through a
proper combination of three universal constants of physics c, G and ~ a set of basic dimensionful
quantities (named the Planck length, time and energy) could be obtained. These quantities have
been given a more precise physical meaning by identifying them as those at which quantum
gravity effects, that in other contexts are of no importance,
are turned on and should become
q
~G
observable. It is then the Planck length, given by lP =
= 1.62 × 10−33 cm, that we expect
c3
to play a role in setting the limits inside of which the quantization of spacetime ought to reveal
itself. The assumption of the existence of a minimal observable length (usefully presented in
[22]) is related to gravity since resolving smaller and smaller distances would require the use of
wavelengths becoming ever shorter (and frequencies ever higher), and since E = hν this means
large energies which at some point start modifying spacetime through their gravitational effects
as it is easily figured out due to the well-known energy-mass equivalence. The interesting thing is
that by following this path gravity would not be an obstacle for the normalizability of quantum
field theories anymore, becoming instead the very element to guarantee such a regularization
by means of the introduction of a proper cutoff occurring at ultraviolet
lengths (that is, at high
q
~c5
19
energies of magnitude comparable to the Planck energy EP =
G = 1.22 × 10 GeV). Fur-
thermore, not only gravitational arguments (see for example [8]) but also string-theoretical ([3],
[13]), cosmological and black-holes ones ([1],[18]) as well as those descending from noncommutative geometries and field theories ([12], [23]) have been shown to coherently point towards the
1
2
existence of a minimal observable length.
Anyway, such an assumption clearly must come to grips with ordinary quantum mechanics governed by Heisenberg’s uncertainty principle. The latter stating that in a whatsoever physical
system the uncertainties in the measurements of the conjugate variables position - linear momentum obey the relation ∆x∆p ≥ ~2 , there is no theoretical limit of any kind on the precision
with which any of the two variables can be determined (in an ideal measurement) provided that
the indeterminacy on the other variable is let go to infinity. The introduction of a minimal length
as pointed out in the foregoing discussion is then to be naturally related to a modification of
Heisenberg’s commutation relations between the operators x̂ and p̂ which introduces new interesting and nontrivial features affecting the ordinary quantum-mechanical context, as we will
show following the treatment in [11]. The new framework originating from what we have outlined
goes under the name of generalized uncertainty principle (GUP). Exploring this landscape appears promising as experimental tests probing the behaviour of the canonical commutator [x̂, p̂]
at a scale as small as that of the Planck length are currently being conceived and investigated,
as the one using quantum optics recently developed in [19]; this happens in a scenery where the
direct access to energies as high as EP keeps being far outside the reach of the most powerful
accelerators of the present era, the maximum scattering energies attainable nowadays and in the
next years at LHC being of the order of 10 Tev.
This said, the subject on which we focus in our work consists of investigating the effects of
the GUP when brought into the problem of relativistic electrons moving in a two-dimensional
space when an external static uniform magnetic field is applied; the starting point is then the
fundamental equation of relativistic quantum mechanics, i.e. the 2+1 dimensional (in our case)
Dirac equation. Solving of this problem within ordinary relativistic quantum mechanics was
first accomplished by Rabi in 1928 ([20]), giving a spectrum for the energy eigenvalues which
is the relativistic generalization of the well-known Landau levels one finds in the domain of
the nonrelativistic theory. Here we will show that even in the context of the GUP the problem
remains exactly solvable; this has been seen to be the case for various other systems, like the
one-dimensional harmonic oscillator studied in [11], the multidimensional one solved in [6] and
the nonrelativistic limit (making use of the Pauli hamiltonian) of the present problem analysed
in [16], but the case dealt with in this work has never been studied in detail to our knowledge,
at least in the present terms.
Moreover, it is certainly worth saying that the study of electronic 2D systems like the one being
the object of this work may find important experimental applications and direct tests in the
scenery set by materials such as the recently isolated graphene [9], for which A. Geim and K.
Novoselov have been awared with the Nobel Prize in Physics in 2010 and in which electrons manifest unique properties requiring them to be treated as highly-relativistic fermions as it nowhere
else happens; one may consult [5] in this regard. In consequence of this, electronic transport
properties of graphene also make it a rightful candidate to come up to (if not to substitute,
under some respects and in a hopefully not so far future) the wide used silicon.
The strategy we adopt in going through the calculations makes use of the momentum space
representation, which is preferable to adopt in minimal length models, and relies on the choice
of the symmetric gauge for the vector potential, which together with the properties of polar co-
3
ordinates allows us to completely separate the variables of the problem and reduct ourselves to
a system of “radial” (in momentum space) differential equations. By means of a suitable change
of variable, these equations can be cast into a form of the Schrödinger type in presence of a
Pöschl-Teller potential hole, whose solution is known. In order to better understand both the
physics and the mathematics which emerge from our treatment, we first check whether and how
our approach is able to reproduce the expected results for the no-GUP case.
At first we shall present the Dirac equation and its main properties. Then we will develop the
algebraic framework underlying the GUP, for what it’s necessary for our problem and slightly
beyond. After this, we will concentrate on the very core of the present work, that is the setting-up
and solving of the Dirac equation, first within ordinary QM and then in the context of minimal
length theories. In the conclusions we have an outlook and discuss the results and their relevance.
Part I
The background
Chapter 1
The Dirac equation
1.1
Beyond the Schrödinger equation
The Schrödinger equation for the wavefunction ψ(x, t) describing a certain physical system (here
a particle of mass m)
∂
i~ ψ(x, t) = Hψ(x, t) =
∂t
~2 ∇2
−
+ V (x, t) ψ(x, t)
2m
(1.1)
is undoubtedly one of the hallmarks of all quantum theory as well as the cornerstone of Schrödinger’s
wave mechanics representation: in this approach to the subject, all what can be told about the
system ultimately descends from this equation. However, since it was first formulated it was
clear it could not be the ultimate word on the topic: indeed it is not invariant with respect to
the Lorentz transformations of special relativity (i.e. it has not the same form in every inertial
reference frame, as all law of physics should), while it can easily be shown to be so under a
galilean change of referential. From this is to be inferred that the domain where the Schrödinger
equation holds true is expressed by the condition v ≪ c, where v is the (maximum) speed of
the particles making up the system; this is confirmed from experimental evidence. It is the very
same situation one has when considering Newton’s classical equations of dynamics: this does not
surprise us, since the Schrödinger equation is written taking into account the classical hamiltonian formalism and the correspondence principle (stating that under some specific conditions,
namely those in which large quantum numbers are selected, or in other words when the quantum of action h can be considered negligibly small, the equations of quantum mechanics reduce
to their classical analogue). Thus it is totally reasonable that when the classical equations of
motion cease to be valid, the Schrödinger equation behaves the same. Briefly speaking, it is a
nonrelativistic (NR) equation.
Now, it turns out that a wide range of phenomena in physics cannot be correctly and thoroughly
accounted for without the use of relativity together with quantum mechanics. A simple example
of how these two theories should proceed together is furnished by the hydrogen atom, where
the observed fine structure of the spectral lines could not be explained by exactly solving the
Schrödinger equation; the mean speed of the electron “orbiting” around the nucleus indeed turns
out to be high enough for relativistic effects to come into play. The need for a quantum mechanical equation also capable of matching with the requirement of relativistic covariance was one of
5
6
CHAPTER 1. THE DIRAC EQUATION
the most felt topics in theoretical physics during the second half of the 1920s. Though the first
step in this direction was made by O. Klein and W. Gordon in 1926, it was Dirac who in 1928
succeeded in finding the right answer to the question.
1.2
The Klein-Gordon equation
In the search for a relation describing relativistic electrons, one has to remember that they are
particles whose quantum nature can never be totally put aside, they being endowed with a spin
of value
1
2.
In order to be able to use the correspondence principle, then, it is better to start
with the simpler case of a relativistic spinless particle just to understand what goes on when
bringing special relativity into quantum mechanics.
Since the Schrödinger equation for a free particle of mass m and charge q can be found starting
from its classical kinetic energy
E=
p2
2m
(1.2)
p → −i~∇
(1.3)
by operating the replacements
E → i~
∂
,
∂t
and applying both sides to a wavefunction ψ(x, t), Klein and Gordon suggested trying the same
from the relativistic energy
E=
which leads to
i~
p
m2 c4 + p2 c2
p
∂
ψ(x, t) = m2 c4 − ~2 c2 ∇2 ψ(x, t) .
∂t
(1.4)
(1.5)
The sense of such an expression is rather questionable, not being clear how a differential operator
under square root should act. This difficulty is easily avoided if we instead use the square of
(1.4)
E 2 = m2 c4 + p2 c2
bringing us to
− ~2
∂2
ψ(x, t) = m2 c4 − ~2 c2 ∇2 ψ(x, t)
2
∂t
(1.6)
(1.7)
which now has a much better appearance and is usefully reformulated as
where 2 =
1 ∂2
c2 ∂t2
m2 c2
2+ 2
~
ψ(x, t) = 0
(1.8)
− ∇2 is the d’Alembertian operator. Eq. (1.8) is called the free Klein-Gordon
equation, and we note it is differential of the second-order in time, then requiring us to know not
only ψ but ∂ψ
∂t too at a given instant in order to describe the system at any later time. However,
other problematic matter is behind the corner. Investigation of the solutions gives
i
i
ψ(x, t) = Ae ~ p·x− ~ Et
(1.9)
1.2. THE KLEIN-GORDON EQUATION
7
along with the dispersion relation
E=±
p
m2 c4 + p2 c2
(1.10)
that is different from (1.4): we have introduced negative energy solutions (also called negative
mass ones) which cannot be simply discarded if we want a complete set of solutions of (1.8).
For a free particle, the spaces of negative and positive energy solutions must be radically distinguished as there’s no physical way for particle to pass on from positive to negative energies
and vice-versa; it follows that the latter do not have a real physical significance (at least in this
context; ultimately the will be of great interest) and are thus called “parasitic” solutions.
If we add an electromagnetic field described by the scalar potential φ(x, t) and the vector potential A(x, t), we know that in the classical hamiltonian treatment it is possible to account for
the presence of this field through the replacements
q
p→p− A ,
c
thus modifying (1.4) to
r
E → E − qφ
(1.11)
q
m2 c4 + (p − A)2 c2 .
c
(1.12)
q 2
(E − qφ)2 − p − A c2 = m2 c4
c
(1.13)
E = qφ +
Then acting the same as above
with consequent negative energies, and (1.8) becomes now
"
#
2 ∂
q 2 2
i~ − qφ − −i~∇ − A c ψ(x, t) = m2 c4 ψ(x, t)
∂t
c
(1.14)
that is the Klein-Gordon equation in presence of an electromagnetic field. It is commonly written
using a different notation to underline its invariance properties; in fact, in terms of four-vectors
in Minkoski spacetime (Greek superscripts and subscripts range from 0 to 3, Latin ones from 1
to 3)
Aµ = (φ, A)
1 ∂
∂
≡ ∂µ = (
, ∇)
µ
∂x
c ∂t
(1.15)
and also, with the metric tensor defined as
gµν = diag(1, −1, −1, −1)
g µν gν̺ = δµ̺
(1.16)
permitting us to raise and lower indexes (and using Einstein’s indexes convention for summations)
Aµ = gµν Aν = (φ, −A)
∂ µ = gµν ∂ν
(1.17)
eq. (1.14) can be restated as
q
q µ m2 c2
µ
∂µ + i Aµ ∂ + i A + 2 ψ = 0
~c
~c
~
(1.18)
8
involving only scalar quantities and then automatically having the same form in every inertial
reference frame.
The major shortcoming with the Klein-Gordon equation when one wants to use it as the equation
describing the motion of relativistic quantum particles of spin 0 is that it doesn’t lend itself to
a statistical interpretation of the wavefunction ψ in the way the Schrödinger equation does. In
fact, for the latter the standard procedure is to define ̺(x, t) = ψ ∗ ψ as the (positive-definite)
probability density of finding the particle in x at a time t; then from (1.1) we easily deduce
∂̺
∂
∂
i~ ∗ 2
i~ ∗
= ψ∗ ψ + ψ ψ∗ =
[ψ ∇ψ − ψ∇ψ ∗ ]
ψ ∇ ψ − ψ∇2 ψ ∗ = ∇ ·
∂t
∂t
∂t
2m
2m
and letting
i~ ∗
[ψ ∇ψ − ψ∇ψ ∗ ] = ℜ
J =−
2m
~ ∗
ψ ∇ψ
im
(1.19)
(1.20)
we find the continuity equation for the probability current density j
∂̺
+∇·j =0 .
∂t
(1.21)
indicating the conservation of probability. Using the four-vector j µ = (c̺, j), it can also be
written
∂µ j µ = 0 .
(1.22)
Following the same path for the free-particle Klein-Gordon equation (1.8), we find
m2 c2 ∗
ψ ψ
~2
m2 c2
ψ2ψ ∗ = − 2 ψψ ∗
~
ψ ∗ 2ψ = −
(1.23)
(1.24)
and subtracting
ψ ∗ 2ψ − ψ2ψ ∗ = ∂µ (ψ ∗ ∂ µ ψ − ψ∂ µ ψ ∗ ) = 0 .
(1.25)
Thus the continuity equation in this case can be satisfied taking
µ
jKG
= α (ψ ∗ ∂ µ ψ − ψ∂ µ ψ ∗ )
(1.26)
with α being a constant we can choose in order to have
and
jKG → j
(1.27)
0
jKG
≡ ̺KG → ̺
c
(1.28)
in the NR limit, which consists in the particle energy reducing to the rest energy, i.e.
i~
∂ψ
≃ mc2 ψ .
∂t
(1.29)
1.3. DIRAC’S SOLUTION
One finds in this limit
̺KG
9
α
= 2
c
∂ψ ∗
−ψ
ψ
∂t
∂t
∗ ∂ψ
→
2M α ∗
ψ ψ
i~
(1.30)
i~
so that α = 2m
gets the work done, even for the fulfilling of (1.27). So we have found for the
free Klein-Gordon equation
∂ψ ∗
i~
∗ ∂ψ
−ψ
ψ
,
(1.31)
̺KG =
2m
∂t
∂t
which is not positive-definite, thus preventing its use as a probability density like it was for
̺ = ψ ∗ ψ. As Pauli and Weisskopf have indeed shown, (1.8) correctly describes a spin-0 scalar
field, with (1.31) being proportional to the electric charge density and (1.25) indicating charge
conservation. The fact that the latter occurs, and not particle conservation as in the Schrödinger
theory, appears reasonable if one thinks that within relativistic domain particles can be created
and annihilated, their mass being converted to energy (the true conserved quantity) through
E = mc2 . Nevertheless, better a try can be made to keep the probabilistic interpretation alive
even in this context.
1.3
Dirac’s solution
As mentioned, the attempt to discover a suitable equation for spin- 12 fermions had to be successfully undertaken by the Englishman P. Dirac. We can keep working bearing in mind the analogy
with the NR theory: as it is known, in this case the introduction of spin renders it necessary to
turn to a two-dimensional vectorial wavefunction called “spinor”
ΨN R


ψ (+)

=
(−)
ψ
(1.32)
and the crucial point of Dirac’s reasoning links to this fact, as he understood that the way to
a coherent relativistic formulation was the use of vectors and matrices. We then think about a
wavefunction made up of several components

ψ1 (x, t)





 ψ2 (x, t) 


Ψ=

..


.


ψN (x, t)
(1.33)
with a position probability density given by the formula
̺(x, t) ≡
N
X
s=1
|ψs |2 .
(1.34)
If we require Ψ to completely specificate the dynamical state of the electron at a certain time,
we will then be searching for a differential equation of first order with respect to time like
10
Schrödinger’s one; for similarity, we write
i~
∂Ψ
= HD Ψ
∂t
(1.35)
with the hermiticity of HD imposed for the same reasons applying to the NR quantum theory,
namely those of probability conservation. Furthermore, the request of invariance under Lorentz
transformations together with formal simmetry of the latter between temporal and spatial coordinates can be taken as a hint to try with a differential equation that is first-order both in time
and space; the validity of this hypothesis will be soon checked.
With no external field, the traslational invariance that must hold as well as all the assumptions
hitherto made can be summarized by the simple general scripture
HD = cα · p + βmc2
(1.36)
with p = −i~∇ as usual (traslation generator) and where β and
α = (αx , αy , αz )
(1.37)
are hermitean operators acting only on the spin space. Eq. (1.35) has become
∂
2
i~ − cα · p − βmc Ψ = 0
∂t
(1.38)
and what we may now do is multiplying on the left by the conjugate operator (meant in fourvectorial sense) to get the scalar equation
∂
∂
2
2
i~ + cα · p + βmc
i~ − cα · p − βmc Ψ = 0
∂t
∂t
2
imc i
m2 c2 2
1 ∂
i j
i
α β + βα ∂i + 2 β Ψ = 0
=⇒ 2 2 − α α ∂i ∂j −
c ∂t
~
~
(1.39)
that should match with (1.8) since both are second order equations for a relativistic system.
Therefore we require
αi αj + αj αi = 2δij I
i
(1.40a)
i
(1.40b)
2
(1.40c)
α β + βα = 0
β =I
thus completely specifying the free Dirac equation (1.38). It is to be underlined that the relations
we have just detected and enumerated can only be satisfied if the αi and β are matrices, and not
merely algebraical quantities; this consideration is what ultimately led Dirac on the right path.
Adding an electromagnetic field, all we have to do is to use (1.11), yielding the Dirac equation
for a particle of spin- 21 moving in presence of an external field
q ∂
2
i~ − qφ − cα · p − A − βmc Ψ = 0 .
∂t
c
(1.41)
11
Let us also add a note on the labelling of the Dirac equaton. When referring to its dimensionality,
it is customary to mention the number of involved space coordinates as well as the temporal one
(due to the symmetry there is in relativistic domain), so one speaks of the (n + 1) Dirac equation
with n = 1, 2, 3 giving the coordinate variables and the 1 standing for the time dimension.
1.3.1
Dirac matrices
It is worth pointing out that the Dirac equation has, like the Klein-Gordon one and for the same
reasons, negative energy solutions: this is a fact one always has to bear in mind, and further
investigation on it ultimately brought Dirac to predict the existence of positrons, discovered in
1932.
In a two-dimensional physical space, we obviously have only two nonzero components of p and
consequently only two of the three α matrices in (1.37) figure in the Dirac equation: with β, in
total three matrices are needed. Even more simply, working on a monodimensional space only
calls for use of two matrices. Looking at (1.40) and bearing in mind the hermiticity demand as
well, it is easy to see that the three Pauli matrices σ = (σx , σy , σz ) of the spin- 12 NR theory
σx =
!
0 1
1 0
σy =
!
0 −i
i
0
σz =
1
0
!
0 −1
(1.42)
are a satisfactory choice. Since they are also the only set of 2 × 2 linearly-independent anti-
commuting matrices, this choice is also unique in the 2 × 2 space and no further discussion is
required as long as we limit ourselves to mono or bidimensional problems.
In a 3D physical space things are quite different, all four matrices α and β having to be used and
the (2 × 2) space to be abandoned. From (1.40a) and (1.40c) the eigenvalues of these matrices
can only be ±1; furthermore (1.40b) with the aid of (1.40a), (1.40c) and the invariance of the
trace under cyclic permutations implies
αi = −βαi β
β = −αi βαi
(1.43a)
(1.43b)
(1.43c)
Tr(αi ) = −Tr(β 2 αi ) = −Tr(αi )
Tr(β) = −Tr[(αi )2 β)] = −Tr(β)
(1.44a)
(1.44b)
so all matrices have to be traceless. From these two facts together is straightforward to realize
that the dimension of the α and β must be an even number. Thus 4 × 4 matrices are the next
possible choice, though here we will not dip into their calculation; nevertheless, one can show
that the dimensionality of the spin space remains 2, thus assuring that we are dealing with
fermions of spin 12 .
12
1.3.2
Probability current density
The wavefunction column vector (1.33) will, most generally, have four components:

ψ1

 
 
ψ2 

Ψ=
  .
ψ3 
 
ψ4
(1.45)
Let us denote its Hermitean conjugate as
Ψ† = ψ1∗ ψ2∗ ψ3∗ ψ4∗
(1.46)
so that (1.34) is written by means of the scalar product
̺(x, t) = (Ψ† Ψ) .
(1.47)


3
X
q
∂Ψ 
αj −i~∂j − Aj + βmc2  Ψ
= qφ + c
i~
∂t
c
(1.48)
If Ψ is a solution of (1.41),
j=1
so that taking the complex conjugate
X
q ∂Ψ†
−i~∂j + Aj Ψ† αj − mc2 Ψ† β
i~
= −qφΨ† + c
∂t
c
3
(1.49)
j=1
and scalar multiplying (1.48) on the left by Ψ† , (1.49) on the right by Ψ and then adding term
by term leaves us with
i~
X ∂̺
∂
(Ψ† Ψ) ≡ i~
= −i~c
∂j Ψ† αj Ψ .
∂t
∂t
(1.50)
j
Recognition of this last expression being the continuity equation for the probability density
allows one to identify
j(x, t) ≡ c Ψ† αj Ψ
(1.51)
so that the statistical interpretation is safe in the Dirac theory.
1.3.3
Covariant formulation
We conclude this chapter presenting another formulation of the Dirac equation, the so-called covariant one, widely used when symmetry and relativistic invariance properties are of importance.
We put
γ µ ≡ (γ 0 , γ)
γ0 ≡ β
γ ≡ βα
(1.52)
13
then by multiplication of (1.41) on the left by β one has the compact expression
h
i
q cγ µ i~∂µ − Aµ − mc2 Ψ = 0
c
(1.53)
known as the covariant Dirac equation. The properties of the γ µ follow from (1.40) and from
the hermitean conditions, and read
γ0γ0 = β2 = I
(1.54a)
γ 0 γ j + γ j γ 0 = β 2 αj + βαj β = αj − αj = 0
(1.54b)
γ i γ j + γ j γ i = βαi βαj + βαj βαi = −β 2 αi αj − β 2 αj αi = −(αi αj + αj αi ) = −2δij I
(1.54c)
(γ 0 )† = γ 0
(1.55a)
(γ j )† = (βαj )† = αj β = −βαj = −γ j
(1.55b)
with the possibility for them to be condensed onto
γ µ γ ν + γ ν γ µ = 2gµν
(1.56)
µ †
(1.57)
(γ )
0 µ 0
= γ γ γ .
Conjugating (1.53) gives
q c −i~∂µ − Aµ Ψ† (γ µ )† − mc2 Ψ† = 0
c
(1.58)
and if we multiplicate on the right by γ 0 and make the position
Ψ ≡ Ψ† γ 0 =⇒ Ψ† = Ψγ 0
(1.59)
where Ψ is called the adjoint of Ψ, with the aid of (1.57) we can restate (1.58) as
q c −i~∂µ − Aµ Ψγ µ − mc2 Ψ̄ = 0 .
c
(1.60)
Clearly we are just going over the steps (1.48-1.51) again with a different notation; the last move
is to perform the scalar multiplications of (1.53) on the left by Ψ and of (1.60) on the right by
Ψ, and then subtracting we get
i~c∂µ Ψγ µ Ψ = 0
(1.61)
that is an invariant relation for the conservation of the probability, i.e. the continuity equation
∂µ j µ = 0
with
j µ ≡ c Ψγ µ Ψ
(1.62)
being no other than j µ = (c̺, j) as it is easily verified from (1.47),(1.51) and (1.52).
(1.63)
Chapter 2
Generalized uncertainty principle
2.1
2.1.1
Heisenberg’s uncertainty relation
Uncertainty in ordinary quantum mechanics
When the theory of quantum mechanics was developed, many thoughts, beliefs and assumptions
of what is now named classical physics, which had up to then exhaustively explained a wide and
various range of phenomena though, were found to be integrated and revisited if not discarded.
One of the most important and founding ones to undergo such a fate was the idea that physical
quantities could, under ideal measurements (suitable instruments, no systematic errors etc...),
always be known with as much precision as one wished. In the quantum theory of Schrödinger
and Heisenberg, much more care is to be taken when formulating these kind of statements:
in fact, it makes a lot of difference whether we are talking about measurements of a single
physical quantity or more of them are instead to be contemporarily known. While in the first
case ordinary quantum mechanics does not pose any formal limit on our capability to gain as
much information we could possibily want from a measurement (but we will see things change
later), in the second things turn out to be rather different: for certain couple or sets of physical
observables of a given physical system, simultaneous measurements cannot be as precise as one
wishes, their uncertainties obeying a well-defined law.
Both in Schrödinger’s wave mechanics approach, by means of the properties of the Fourier
transforms, and in Heisenberg’s matrix mechanics using the commutator
[x̂, p̂] = i~
(2.1)
one shows that for the canonical observables x (space coordinate, for simplicity assuming a onedimensional system) and p (linear momentum) when performing contemporary measurements
of both variables the relation
∆x∆p ≥
14
~
.
2
(2.2)
2.1. HEISENBERG’S UNCERTAINTY RELATION
15
holds, ∆x and ∆p being the rms of the measurements, defined as1
∆x ≡
∆p ≡
p
p
hx2 i − hxi2
(2.3)
hp2 i − hpi2 .
(2.4)
Eq. (2.1) descends from the representations of the canonical operators
x̂ψ(x) = xψ(x)
x̂ϕ(p) = i~
∂ϕ(p)
∂p
p̂ψ(x) = −i~
∂ψ(x)
∂x
p̂ϕ(p) = pϕ(p)
(2.5)
(2.6)
respectively on coordinate and momentum spaces; (2.2) is simply generalizable to higher-dimensional
spaces, being true for every couple of cartesian conjugate variables (x and px , y and py , z and
pz in a 3D space). As shown in Figure 2.1, (2.2) is equivalent to stating that we can indefinitely
Figure 2.1: Representation of Heisenberg’s uncertainty relation. The allowed region is at the
right of the curve.
decrease the undeterminacy on one of the two variables at the price of progressively losing all
the information on the other. There is a forbidden zone between the axes and the line which
was completely absent in the classical theory, where the quantum of action ~ is considered to be
zero.
2.1.2
Position and momentum eigenstates
It follows from what we have said that in ordinary quantum mechanics one can righteously search
for states with a definite value of the position coordinate or momentum, that is, eigenstates of
the position or momentum operators with given eigenvalues. Adopting the standard bra-ket
1
hai denotes the mean value of a in a certain state.
16
CHAPTER 2. GENERALIZED UNCERTAINTY PRINCIPLE
notation,
x̂ |x̄i = x̄ |x̄i
x̄ ∈ R
(2.7)
p̂ |p̄i = p̄ |p̄i
p̄ ∈ R
(2.8)
is what we ask, where clearly ∆x|x̄i = ∆p|p̄,ni = 0. Solutions are readily found using (2.5) and
(2.6), and they are nothing else than plane waves or Dirac-delta functions (depending on the
chosen representation), e.g.
ψx̄ (x) ≡ hx| x̄i = δ(x − x̄)
i
1
ψp̄ (x) ≡ hx| p̄i = √
e ~ p̄x
2π~
(2.9)
projecting the eigenstates in the x space and
ϕp̄ (p) ≡ hp| p̄i = δ(p − p̄)
i
1
e− ~ px̄
ϕx̄ (p) ≡ hp| x̄i = √
2π~
(2.10)
projecting them in momentum space. One passes from one space to the other using the Fourier
transform and anti-transform. We also note that by varying the localization parameters x̄ and
p̄ one obtains the orthonormal complete sets {|x̄i}x̄∈R and {|p̄i}p̄∈R of position and momentum
eigenstates. Now the concern with these eigenstates may be that they are a mathematical solution
that lacks real physical meaning, because their normalization integrals2
Z
2
|ψx̄ (x)| dx =
Z
2
|φx̄ (p)| dp ,
Z
2
|ψp̄ (x)| dx =
Z
|φp̄ (p)|2 dp
(2.11)
diverge: having perfectly monochromatic waves is not feasible in practice. However, in the present
context this is only a formal difficulty and can be easily surmounted considering that the eigenstates (2.7) and (2.8) can be approximated with arbitrary precision by the sequence of finiteenergy normalizable states |x̄, ni and |p̄, ni with ever decreasing (but finite) uncertainties
|x̄i = lim |x̄, ni
n→∞
n→∞
|p̄i = lim |p̄, ni
n→∞
n→∞
lim ∆x|x̄,ni = 0
(2.12)
lim ∆p|p̄,ni = 0
(2.13)
an observation that can someway be compared to the physical way of thinking at an ideal
monochromatic wave as the result of the limiting process of narrowing a wave packet. Obviously,
(2.2) does not allow an eigenstate to verify both (2.7) and (2.8), that is to have at the same time
a definite position and momentum, as we have already discussed.
All what has been said here can be directly extended to spaces with more than one dimension,
being valid for each of the Cartesian directions (e.g. x, y, z in a 3D space) and remembering
that
2
[x̂i , x̂j ] = [p̂i , p̂j ] = 0
(2.14)
[x̂i , p̂j ] = i~δij
(2.15)
The equalities hold thanks to Parseval’s theorem.
2.2. MINIMAL LENGTH MODELS
17
so that positions and momenta along different axes can be measured at the same time with no
precision limitations. For example, with obvious notation,
2.2
2.2.1
hx| x̄i = ψx̄ (x) = δ3 (x − x̄)
hx| p̄i = ψp̄ (x) =
hp| p̄i = ϕp̄ (p) = δ3 (p − p̄)
hp| x̄i = ϕx̄ (p) =
1
(2π~)
1
(2π~)
i
3
2
e− ~ p̄·x
i
3
2
e− ~ p·x̄.
(2.16)
(2.17)
Minimal length models
A modified inequality
We now focus on the theoretical structure that is of interest for the present work, i.e. the
generalized uncertainty principle (GUP), closely following [11]. Under this label models are
developed in which Heisenberg’s relation (2.2) is shifted to
∆x∆p ≥
~
1 + ω(∆x)2 + η(∆p)2 + τ
2
(2.18)
where ω, η and τ are (small) positive deformation parameters whose magnitude and onset is to
be related to quantum gravity effects and that may, in general, also depend on the expectation
values of x and p. Throughout the present work we will focus on a special case of the GUP, e.g.
one having (only) a minimal length and specified by
ω=0.
(2.19)
The above restriction indeed has the effect of rendering the p-space more similar to that of the
ordinary QM case than that of position, in the sense that no minimal uncertainty in momentum
will exist, as we will se in what follows. For this reason, working in momentum space has
undoubtedly some advantages and this will be our choice when we come up to the solving of the
problem to which this work is ultimately devoted.
It is instructive to plot the minimal uncertainty curve in order to visualize the relationship
between the uncertainties on the two observables, since it is not as simple as in ordinary quantum
mechanics. One sees in Figure 2.2 that when increasing ∆p a point is encountered where the
curve reaches its minimum ∆x value and then steers back, which is something totally new and
different from the branch of hyperbola we met in Figure 2.1: the uncertainty on the position
variable can no longer be made arbitrarily small but instead has a lower bound which we will
call ∆x0 . This is a remarkable fact whose physical consequences are not at all trivial, and the
mathematical-algebraical context in which it takes place is worth a detailed study to which
we will now dedicate ourselves. Here it can also be observed that (2.19) has set the “minimal
measurable momentum” ∆p0 = 0, while in more general and complex GUP frameworks this does
not happen.
18
Figure 2.2: Modified uncertainty relation. Physically allowed zone on the right.
2.2.2
Minimal length algebra in one dimension
We first demonstrate the following general result:
Theorem. Let A and B be a pair of observables represented by symmetric operators
domain of
Â2
and
B̂ 2 .
3
on a
Then
∆A∆B ≥
|h[Â, B̂]i|
.
2
(2.20)
Proof. We introduce the new observables
which immediately verify
e ≡ A − hAi
A
e ≡ B − hBi
,B
be be
A,
B = [Â, B̂] .
(2.21)
(2.22)
(to simplify the notation, from now on we do not write the operator symbolˆ). Also, by definition
of rms deviations, one has
q
q
p
e ≡ hA
e2 i − hAi
e 2 = hA
e2 i = hA2 i − hAi2 = ∆A
∆A
q
q
p
e ≡ hB
e 2 i − hBi
e 2 = hB
e 2 i = hB 2 i − hBi2 = ∆B.
∆B
(2.23)
(2.24)
Let now the state of the system be described by the normalized ket |ui contained in DA and
3
An operator Ô is said to be symmetric on domain DO if
1. DO is dense
2. (hφ| Ô) |ψi = hφ| (Ô |ψi) ∀ |φi , |ψi ∈ DO .
From this definition it follows that hφ| Ô |φi ∈ R for any |φi ∈ DO .
19
DB ; then
e 2 (∆B)
e 2 ≡ hu| A
e2 |ui hu| B
e 2 |ui
(∆A)2 (∆B)2 = (∆A)
eA
e |ui) hu| B(
e B
e |ui = hu| A)
e
e |ui hu| B
e B
e |ui
= hu| A(
A
(2.25)
where the last equality is a consequence of the symmetry of the observables. Applying the
Schwarz inequality4 , (2.25) leads to
eB
e |ui |2 .
(∆A)2 (∆B)2 ≥ | hu| A
Then decomposing
(2.26)
ee e e
ee e e
e e
ee e e
eB
e = AB + B A + AB − B A = AB + B A + [A, B]
A
2
2
2
2
(2.27)
renders it possible to write
eB
e |ui =
hu| A
so that (2.26) becomes
*
eB
e+B
eA
e
A
2
+
+
*
e B]
e
[A,
2
+
*
+ *
+2
A
e B]
e e+B
eA
e
[A,
eB
+
(∆A) (∆B) ≥ .
2
2
2
2
(2.28)
(2.29)
It is straightforward to prove that, exactly as it is done when self-adjointness of Â and B̂ holds,
eB+
e B
eA
e
A
e and B
e are so; its expectation values are then real (cfr.
is a symmetric operator since A
2
footnote on page 18). On the contrary,
eB−
e B
eA
e
A
2
will be antisymmetric, with purely imaginary
diagonal elements. We can thus conclude, using (2.22) as well,
(∆A)2 (∆B)2 ≥
*
eB
e+B
eA
e
A
2
+2
+
e B]i|
e 2
e B]i|
e 2
|h[A,
|h[A,
|h[A, B]i|2
≥
=
.
4
4
4
(2.30)
and extracting the square root the proof is complete.
With this in mind, we can investigate what kind of commutation relation between the canonical observables is to be associated to (2.18), assuming them to be symmetric but not necessarily
hermitian for reasons we shall later mention. The answer is easily found out being
[x̂, p̂] = i~(1 + ωx2 + ηp2 )
(2.31)
provided one sets τ = ωhxi2 + ηhpi2 . Sticking to (2.19), we then rewrite for the sake of clarity
4
This is an elementary result of linear algebra, stating that hφ|φi hψ|ψi ≥ | hφ|ψi |2 .
20
our basic relations
~
1 + η(∆p)2 + ηhpi2
2
[x̂, p̂] = i~(1 + ηp2 )
∆x∆p ≥
(2.32)
(2.33)
which we will assume true in all what follows.
Up to now we have only qualitatively shown the existence of ∆x0 ; its determination is simply
about starting from the minimal uncertainty equality obtained from (2.32), solving for ∆x,
differentiating with respect to ∆p, equating to zero to find
∆p =
s
1 + ηhpi2
η
(2.34)
(momentum uncertainty corresponding to minimal position uncertainty) and finally reinserting
in the initial equation to get
√ p
∆xmin (hpi) = ~ η 1 + ηhpi2 .
(2.35)
hpi = 0
(2.36)
√
∆x0 = ~ η .
(2.37)
The smallest uncertainty in position measurements is obviously reached when
and reads
The next step is to determine the form of the canonical operators in order to respect (2.33).
Working in momentum space due to its simpler features as already mentioned, the task is easily
worked out by slightly changing (2.6) to
x̂ϕ(p) = i~(1 + ηp2 )∂p ϕ(p)
(2.38)
p̂ϕ(p) = pϕ(p)
(2.39)
as checked with a straightforward calculation. Study of the form assumed by these operators,
particularly by x̂ in (2.38), reveals that requests of symmetry are not satisfied unless some
modifications are brought into the usual Hilbert space of quantum mechanics. In particular, the
measure of integration must be accordingly changed in such a way that
hφ|ϕi ≡
Z
+∞
−∞
dp
φ∗ (p)ψ(p)
1 + ηp2
(2.40)
21
because this guarantees that (performing a partial integration)
hφ| (x̂ |ϕi) =
Z
+∞
−∞
=−
=
Z
Z
dp
φ∗ (p)i~(1 + ηp2 )∂p ϕ(p)
1 + ηp2
+∞
i~ [∂p φ∗ (p)] ϕ(p)dp
−∞
+∞
−∞
∗
dp i~(1 + ηp2 )∂p φ ϕ(p) = (hφ| x̂) |ϕi
2
1 + ηp
(2.41)
where the role of the (1 + ηp2 )−1 measure factor is evidently that of cancelling out with the
one in (2.38). For p̂ not only the symmetry is immediate, but we are exactly retaining its
standard expression of (2.6), the hermiticity thus being guaranteed. This latter property is no
more possessed by x̂ as a rigorous analysis can show; this operator has instead a one-parameter
family of self-adjoint extensions, but this goes beyond the scopes of our treatment of the subject.
Anyway, while the study of momentum operator and its eigenbasis can be carried out exactly
in the same manner, and the same conclusions can be drawn with respect to what has been
previously done [starting from (2.8), obtaining the first of (2.10) and then (2.13)], the new form
of the position operator deserves a closer look which it is interesting as well as necessary to have.
2.2.3
Gaining information on position. Maximally localized states.
The first thing that probably comes to one’s mind when caring for a functional analysis of the
x̂ operator as in (2.38) is attempting to solve its eigenvalue problem, e.g.
i~(1 + ηp2 )∂p ϕa (p) = aϕa (p)
a∈R
(2.42)
and this is something that can be done with ease, separating the differential equation and
integrating to get, up to a constant B,
a
√
ϕa (p) = C exp −i √ arctan( ηp) .
~ η
(2.43)
Normalizing the eigensolution accordingly to the new measure introduced in the previous section
yields
r√
η
dp
2 π
= |C| √ =⇒ C =
1 = CC
2
1
+
ηp
η
π
−∞
r√
η
a
√
ϕa (p) =
exp −i √ arctan( ηp)
π
~ η
∗
Z
+∞
(2.44)
(2.45)
so that the question seems formally solved, and in GUP with no minimal uncertainty in momenta
the positional operator can in fact still be diagonalized by a one-parameter family of eigenstates.
22
To this end the orthogonality between formal eigenstates may be tested:
√ Z +∞
η
dp
a − a′
√
exp −i √ arctan( ηp)
hϕa′ |ϕa i =
π −∞ 1 + ηp2
~ η
+∞
√
~ η
a − a′
√
exp −i √ arctan( ηp) =−
i(a − a′ )π
~ η
−∞
√
2~ η
a − a′
=
sin
√ π
π(a − a′ )
2~ η
and picking the set
E
{ϕ(2n+a)~√η , n ∈ Z}
−1≤a<1
(2.46)
(2.47)
one effectively fulfils the orthonormality relation
D
E
ϕ(2n+a)~√η ϕ(2n′ +a)~√η = δnn′
(2.48)
once the parameter a has been chosen. These facts taken alone could seem to suggest that the
distinction with the ordinary QM structure is that here we have a “lattice” of states in the
coordinate space and apart from this the same things are going on in the two cases, with the
consequence of (superficially) believing that all what one has to do is replacing the set formed
by the second of (2.10) with the one in (2.47). However, we are perfectly aware that these
eigensolutions just found can neither represent physical states, nor can they be approximated
by some kind of physically-sensed |a, ni succession having ∆x|a,ni → 0 for n → ∞ because the
restriction
∆x|a,ni ≥ ∆x0 > 0
(2.49)
always holds. In other words, since all the states with position uncertainty within the “forbidden
gap” [0, ∆x0 [ are now to be excluded, the eigenfunctions (2.45) cannot be approached and are
now really only formal mathematical solutions without physical consistence. In fact they don’t
belong to the domain of the operator p̂ as can be seen from
Z
2 √η +∞ dp
hp i = ϕa p ϕa =
p2
π −∞ 1 + ηp2
2
(2.50)
which diverges, causing the uncertainty in momentum to be infinite. Moreover, if we are to study
2
p
a classical particle the energy (E = h 2m
i) would be infinite for formal position eigenstates, a
clear signal that we can make no use of them; a similar argument also applies to relativistic
systems where we can use E 2 = m2 c4 + p2 c2 .
Lacking a true position eigenbasis due to finite localizability, the standard route of calculat-
ing, given a state |ψi, the matrix elements hx|ψi with their usual direct physical interpretation
is no more accessible. We then have to revise our way of getting data on position in physical
systems, and once one realizes that (2.49) cannot be escaped the best move to make is to find
those states for which this relation relation becomes an equality, i.e. states with smallest possible
uncertainty in position. We label them ϕx̄M L (M L standing for maximally-localized), meaning
23
we shall have
L M L
ϕM
x̂ ϕx̄
= x̂
x̄
M L
M L
ϕx̄ ∆x ϕx̄
= ∆x0
(2.51)
(2.52)
where the second request also implies (2.36). To find them let us reconsider the proof of (2.20),
with A = x and B = p: asking for the equality to hold, two conditions
to be together
D have E
eB+
e B
eA
e
A
be zero. The
verified, namely that Schwarz’s relation reduce to an equality and that
2
first one is equivalent to requiring the two vectors to be proportional, e.g.
L
L
L
x
e ϕM
= (x̂ − hxi) ϕx̄M L = α (p̂ − hpi) ϕM
= αe
p ϕM
x̄
x̄
x̄
with α being a complex constant to be determined. Unfolding the second constraint,
epe + pex
e M L
1 M L M L M L M L ML x
ϕx̄ x
epe ϕx̄
ϕ
+ ϕx̄ pex
e ϕx̄
=
ϕx̄ x̄
2
2
L 2 M L
1
pe ϕx̄
=0
= (α∗ + α) ϕM
x̄
2
(2.53)
(2.54)
where the symmetry of the operators, (2.53) and its conjugate relation have been used. We
deduce that α is purely imaginary, so we can write
α = −ik
k∈R
(2.55)
(the minus sign is used for convenience) and to exactly determine its value we use
L
L
(x̂ − hxi) ϕM
= −ik (p̂ − hpi) ϕM
x̄
x̄
M L
M L
ϕx̄ (x̂ − hxi) = +ik ϕx̄ (p̂ − hpi)
(2.56)
(2.57)
from which scalar multiplication term by term gives
(∆x)2 = k2 (∆p)2 .
But in the present case also
(∆x)2 (∆p)2 =
|h[x̂, p̂]i|2
4
(2.58)
(2.59)
holds, and we see it must be5
k=
~
~
|h[x̂, p̂]i|
=
(1 + ηhp2 i) =
(1 + ηhpi2 + η(∆p)2 ) .
2(∆p)2
2(∆p)2
2(∆p)2
(2.60)
We can now solve (2.53) in momentum space: dealing with the differential equation
L
ML
i~(1 + ηp2 )∂p − x̄ ϕM
x̄ (p) = −ik (p − hpi) ϕx̄ (p)
(2.61)
p̂]i|
Rigorously, we should consider both roots in k = ± |h[x̂,
. However, choice of the minus sign can be seen
2(∆p)2
having a non-normalizable associated solution.
5
24
does not require much effort, in fact passing through
L
i −ik(p − hpi) + x̄
dϕM
x̄
=−
dp
L
~
1 + ηp2
ϕM
x̄
(2.62)
and then integrating, we arrive to
ix̄
khpi
√
− √ − √
arctan ( ηp)
~ η ~ η
2
2
1+ηhpi +η(∆p)
ix̄
hpi[1 + ηhpi2 + η(∆p)2 ]
√
2 −
2
4η(∆p)
exp − √ −
= A(1 + ηp )
arctan ( ηp) .
√
~ η
2 η(∆p)2
(2.63)
k
L
ϕM
= A(1 + ηp2 )− 2η~ exp
x̄
States of absolutely maximal localization are then obtained using (2.34) and (2.36); taking into
account normalization too6
Z +∞
Z +∞
dp
dp
ML ∗ ML
∗
1=
(ϕx̄ ) ϕx̄ = AA
2
2 2
−∞ 1 + ηp
−∞ (1 + ηp )
(
I
dp
1
AA∗
2iπ
= 2
= AA∗ 2 Res
i 2
i 2
i 2
√
√
√
η
η
(p − η ) (p + η )
(p − η ) (p +
√i )2
η
)
p= √iη
π
= |A|2 √
2 η
(2.64)
so finally we have completely determined
L
ϕM
x̄ (p)
r √
√
x̄ arctan ( ηp)
2 η
2 − 12
(1 + ηp ) exp −i
=
√
π
~ η
(2.65)
which is the form assumed in momentum space by the states maximally localized around a generic
position x̄. In our minimal length model, such states are what most closely resembles the plane
waves in (2.10), someway being their generalization. Concerning this, note that in the η → 0
limit (2.65) tends to the corresponding expression in (2.10) if we disregard the normalization
constants and particularly A going to zero (in fact it has already been pointed out that, in the
ordinary context, position eigenstates are not normalizable, so such constant doesn’t play any
L
fundamental role). A major difference, anyway, exists and lies in the ϕM
x̄ (p) being real physical
(normalizable) states of finite energy, for which (proceeding like in the previous integral)
L 2 M L
p ϕx̄
ϕM
x̄
√ Z
2 η +∞
dp
=
p2
π ∞ (1 + ηp2 )2
√ I
2 η
dp
=
p2
i 2
i 2
√
√
πη 2
(p − η ) (p + η )
(
)
√
2 η
p2
2πi Res
=
πη 2
(p − √iη )2 (p + √iη )2
p= √iη
=
6
1
.
η
To compute the integral, we close the path in the upper complex plane exploiting Jordan’s Lemma.
(2.66)
25
It is also useful to check the mutual relationships between these maximally localized states.
What happens is similar to the case of formal position eigenvectors, the scalar product being
L M L
ϕM
ϕx̄
x̄′
√
√ Z
(x̄ − x̄′ ) arctan ( ηp)
2 η +∞
dp
exp −i
=
√
π ∞ (1 + ηp2 )2
~ η
√ Z π
2 η + 2 du
(x̄ − x̄′ )
1
exp
−i
=
u
√
√
π −π
η 1 + tan2 u
~ η
(2.67)
2
and naming κ =
(x̄−x̄′ )
√
~ η
2
π
=
Z
(x̄−x̄′ )
∆x0 ,
+ π2
− π2
du
(2.67) is evaluated as
e2iu + e−2iu + 2
4
e−iκu =
sin ( κπ
)
2 2
κπ
1 − κ4
2
(2.68)
a function whose behaviour is shown in Figure 2.3. We conclude that orthonormality is attained
only between states maximally localized around positions spaced of 4∆x0 , 6∆x0 , 8∆x0 etc.
L M L
ϕx̄
Figure 2.3: Plot of ϕM
over κ.
x̄′
Armed with knowledge of these states, which permit us to recover as much positional information
as possible in our minimal length context, the ordinary-QM approach of projecting a whatsoever
state along position eigenstates is now generalizable for our purposes. Much in the same way
hx|ψi = ψ(x) originated the position representation, the projections
L
ψi
ψ(x̄) = ϕM
x̄
(2.69)
give rise to the so-called quasiposition representation. Properly, (2.69) furnishes the probability
of the particle under observation to be maximally localized around a certain position x̄, and we
26
expect getting back again to the position representation if we put η = 0. Explicitly, one has
Z
+∞
dp
[ϕ M L (p)]∗ ψ(p)
2 x
1
+
ηp
−∞
r √ Z
√
2 η +∞
x̄ arctan ( ηp)
dp
exp i
=
ψ(p)
√
π −∞ (1 + ηp2 ) 32
~ η
ψ(x) =
(2.70)
and as an example it can be checked that δ-“functions” in the space of the momenta (which are
still perfectly legit in this version of the GUP, p being exactly measurable) have a quasipositional
representation given by (we replace x̄ with x)
r √ Z
√
2 η +∞
x arctan ( ηp)
dp
exp
i
ψp̄ (x) =
δ(p − p̄)
√
π −∞ (1 + ηp2 ) 32
~ η
r √
√
2 η
x arctan ( η p̄)
1
exp
i
=
√
π (1 + η p̄2 ) 32
~ η
(2.71)
that is, still a plane wave, but now verifying the dispersion relation
k=
√
√
arctan ( η p̄)
2π~ η
2π
=
=⇒ λ =
√
√
λ
~ η
arctan ( η p̄)
and also for a nonrelativistic particle, using E =
(2.72)
p̄2
2m
√
2π~ η
√
λ=
arctan ( 2mEη)
√ 2π~ η 2 1
E = tan
.
λ
2mη
(2.73)
(2.74)
From (2.72)-(2.73) the same important feature is seen to emerge once again: in the limit of very
large energies, the shortest achievable wavelength is
√
λ0 = lim λ = 4~ η = 4∆x0
E→+∞
(2.75)
and getting to know something below this scale is simply physically impossible, since the Fourier
analysis of the maximally localized “waves” with definite momentum does not contain any wavelength smaller than λ0 . Another way of stating this is also furnished by (2.74), cause λ0 is seen
to be a critical point related to an “UV catastrophe” and thus represents a border value never
to be reached nor overstepped.
We only mention that a further study of the quasiposition representation is obviously possible;
in particular, continuing to follow the treatment given in [11] as we have hitherto done, one can
find the inverse transformation of (2.70) (from quasiposition wavefunctions to momentum-space
ones) as well as the form assumed by the operators x̂ and p̂ and all the Heisenberg algebra in
the maximal localization function space.
2.2.4
27
Extension to higher dimensional spaces
Dealing with one-dimensional systems is of course useful and instructive under some aspects,
essentially due to mathematical simplicity of the related problems often allowing the latter to be
exactly solved: furthermore, this turns out to be of particular use and interest in the development
of theories involving, like the present one, new characteristic “experimental” elements which have
to be tested in their (more or less direct) predictions. In this respect, study of simple but relevant
cases such as the harmonic oscillator in [11] and [24] has helped and helps gaining some insight
into at least the most immediate consequences of the GUP model. With this said, however, it
must be clear that a mono-dimensional theory alone is not of much use and by application of it
solely one doesn’t reach far in describing nature; generalizing (2.32) and (2.33) is then something
we shall discuss, above all in view of the incoming chapter where we will devote ourselves to
facing a two-dimensional case of the Dirac equation.
At a first glance, the most natural way to generalize (2.33) to n dimensions probabily appears
to be
[x̂i , p̂j ] = i~δij (1 + ηp2 )
(2.76)
with p = (p1 , p2 , ..., pn ). Though this is a good choice (and the one we will make in what follows)
preserving rotational symmetry as it is rather evident from its explicit form, it is far from being
unique. Indeed in the same way as one could think about reasonable variations on (2.33), more
general versions of (2.76) expressible as
[x̂i , p̂j ] = i~(1 + f (p))
(2.77)
(with rotational and translational invariance to be imposed on f (p)) can underlie GUP models
where possibly also a minimal uncertainty in some or all components of the momentum figures.
Anyhow, sticking to (2.76) and also requiring
[p̂i , p̂j ] = 0
(2.78)
allows us to extend (2.38) and (2.39) to the present n-dimensional momentum space as
x̂i ϕ(p) = i~(1 + ηp2 )∂pi ϕ(p)
(2.79)
p̂i ϕ(p) = pi ϕ(p) .
(2.80)
Although these relations closely resemble the corresponding ones in one dimension, a characteristic property which could not be observed in our previous treatment now emerges, that is the
non commutativity of different-directional position operators
[x̂i , x̂j ] = −2~2 η(1 + ηp2 )(pi ∂pj − pj ∂pi ) = 2i~η(p̂i x̂j − p̂j x̂i )
(2.81)
telling us that the geometry of this space is pretty different from the one we are used to handle
in ordinary quantum mechanical questions. The panorama outlined by the n-dimensional GUP
thus looks even richer in noteworthy attributes than it was for simple one-dimensional models;
28
if we define in the standard way
n
X
(∆xk )2
(∆x) ≡ hx i − hxi =
2
2
2
(∆p)2 ≡ hp2 i − hpi2 =
k=1
n
X
(∆pk )2
(2.82)
(2.83)
k=1
then making use of (2.20), (2.78), (2.77) and (2.81) we can besides obtain
~ δij 1 + η(∆p)2 + ηhpi2
2
∆xi ∆xj ≥ ~η|hp̂i x̂j − p̂j x̂i i|
∆xi ∆pj ≥
∆pi ∆pj ≥ 0 .
(2.84)
(2.85)
(2.86)
The geometrical feature which we mentioned above and that is worth stressing here comes out
fairly well from (2.85). In fact it is true that one can compute the minimal uncertainties in each
position coordinate adopting the same strategy seen in the mono dimensional treatment, that is
to say, replacing the ≥ with an equality sign in (2.84) and putting hpk i = 0 ∀k, ∆pk = 0 ∀k 6= i
as suggested by (2.86) (different momentum coordinates can be simultaneously known with no
constraints on uncertainties) to find
∆xi ∆pi =
~
1 + η(∆pi )2
2
(2.87)
from which ∆xi,0 = ∆x0 as in (2.37); nevertheless a state maximally localized around a certain
position x̄ of space could not, in general, have a total standard deviation simply given by
|∆x0 | =
q
√
(∆x1,0 )2 + (∆x2,0 )2 + ... + (∆xn,0 )2 = ∆x0 n
(2.88)
owing to the fact that (2.85) sets an intrinsic limit on simultaneous measurements of a particle’s
different coordinates which is to be taken in consideration as well.
The scalar product which ensures symmetry of the canonical operators is straightforwardly
generalized as
hφ|ϕi =
Z
+∞
−∞
dn p
φ∗ (p)ϕ(p)
1 + ηp2
(2.89)
with the same conclusions about the hermiticity of p̂, whereas x̂ is merely symmetric, being
drawn from more advanced studies on the topic.
We finish with a few considerations on angular momenta, the generators of rotations in the case
n = 3, inside our GUP theory. In ordinary quantum mechanics, it is known that
L̂k = ǫijk x̂i p̂j
(2.90)
L̂k ϕ(p) = i~ǫijk ∂pi pj ϕ(p) = −i~ǫijk pi ∂pj ϕ(p)
(2.91)
or, in p-space,
29
(having renamed dummy indexes) and our aim is to preserve this latter expression even in
presence of the modified operators (2.79) and (2.80) if we still want to regard L̂ as the generators
of finite rotations. In order to achieve this, one replaces (2.90) with
L̂k =
for adding the bounded operator
1
1+ηp2
1
ǫijk x̂i p̂j
1 + ηp2
(2.92)
is certainly an allowed operation with no drawbacks. Here
we want to stress that the explicit action of the angular momentum operators on functions is
not modified by the introduction of the GUP, since we have purposely configured things in such
a way that the factor (1 + ηp2 )−1 disappears. This said, we can restate some of the relationship
encountered above, noticing that (2.81) in three dimensions assumes the form
[x̂i , x̂j ] = −2i~η(pˆj x̂i − p̂i x̂j ) = −2i~η(x̂i p̂j − x̂j p̂i ) = −2i~η(1 + ηp2 )ǫijk L̂k
(2.93)
where we have used the properties of the Levi-Civita symbol:
(1 + ηp2 )ǫijk L̂k = ǫijk ǫlmk x̂l p̂m = (δil δjm − δim δjl )x̂l p̂m = x̂i p̂j − x̂j p̂i .
(2.94)
Immediately also (2.85) is rewritten in the more useful and compact form
∆xi ∆xj ≥ ~η h(1 + ηp2 )ǫijk L̂k i
(2.95)
showing that the non compatibility of the observables x̂i and x̂j is strictly related to the angular
momentum along the direction orthogonal to both. Other commutation relations one may wish
to verify are
[p̂i , L̂j ] = i~ǫijk p̂k
(2.96)
[x̂i , L̂j ] = i~ǫijk x̂k
(2.97)
[L̂i , L̂j ] = i~ǫijk L̂k
(2.98)
there being no difference with the well-known results of standard quantum mechanics.
Part II
Solving the problem
Chapter 3
Relativistic electrons in a static
uniform magnetic field
3.1
Setting up the (2+1) Dirac equation
Relativistic electrons moving on a plane in presence of an external magnetic field B are correctly
studied using the Dirac equation where the scalar potential can be set to zero due to absence of
an electric field, i.e.
φ=0
(3.1)
so that (1.41) becomes, setting also q = −e
i~
∂
e − cα · p + A − βM c2 Ψ = 0
∂t
c
(3.2)
Since the magnetic field is constant in time, the same is true also for the vector potential
A = A(x) and we will then be searching for stationary solutions. The well-known strategy to
get rid of time dependence in these cases is to separate
i
Ψ = ψe− ~ Et
(3.3)
getting the time-indipendent Dirac equation
h
i
e
cα · (p + A) + βM c2 ψ = Eψ
c
(3.4)
with E being the energy of the state. We set up our two-dimensional problem fixing the magnetic
field to be directed along the z-axis
B = B0 k̂
(3.5)
and in consequence of that all the dynamics we wish to study lies on the x-y plane. Dimensionality
of the problem also allows us to utilize a 2 × 2 representation for the three Dirac matrices αx ,
αy and β involved in (3.4); we use the Pauli matrices putting
αx = σx
αy = σy
31
β = σz
(3.6)
32
CHAPTER 3. RELATIVISTIC ELECTRONS IN A MAGNETIC FIELD
and consequently a two-component spinor will describe our system

ψ=
ψ1
ψ2


(3.7)
with ψ1 denoting the spin-up state and ψ2 the spin-down one. If one labels
e e P± = Px ± iPy = px + Ax ± i py + Ay
c
c
eq.(3.4) becomes



M c2
cP+
which is nothing but the system




  ψ1 
 ψ1 

=E

ψ2
ψ2
−M c2
cP−
(3.8)
(3.9)
P− ψ2 = ǫ− ψ1
(3.10)
P+ ψ1 = ǫ+ ψ2
(3.11)
having made the position
ǫ± =
E ± M c2
.
c
(3.12)
It is besides possible and useful to obtain second order independent equations from (3.10) and
(3.11), isolating ψ1 from the first and substituting it in the second, and doing the same for ψ2
from the second into the first. Letting
ǫ+ ǫ− = ǫ− ǫ+ = ǫ2 =
E 2 − m2 c4
c2
(3.13)
the decoupled equations are then
P− P+ ψ1 = ǫ2 ψ1
P+ P− ψ2 = ǫ2 ψ2 .
(3.14)
(3.15)
At this point we also wish to already place ourselves in momentum space in view of what follows.
3.1.1
Choice of the gauge. Polar coordinates.
Determining the intensity and direction of our constant static magnetic field obviously completely specifies the physical situation we are going to investigate. However, there still is some
mathematical freedom which can have big influence on the resolution of (3.4): we haven’t given
the form of A yet. The law relating the magnetic field B to the vector potential A in electromagnetism is
B = ∇ × A =⇒ Bz = B0 = (∇ × A)z = ∂x Ay − ∂y Ax
(3.16)
and it leads to a variety of available options, distinguished between “symmetric” and “asymmetric” gauges. An accurate choice, meaning one related to the method of solution one has in mind
3.1. SETTING UP THE (2+1) DIRAC EQUATION
33
to try, is crucial for making the way easier. We notice that adopting the symmetric gauge
Ax = −
B0
y
2
Ay = +
B0
x
2
(3.17)
in elementary agreement with (3.16), the total angular momentum Jz (see Appendix B) is a
constant of the motion. This is easily seen passing onto polar coordinates, described in Appendix
A, if we define (r is not a spatial coordinate here!):
q


r
=
p2x + p2y

py

 ϑ = arctan
px
and equivalently
(
(3.18a)
(3.18b)
px = r cos ϑ
(3.19a)
py = r sin ϑ .
(3.19b)
In this reference frame (3.17) is spherically symmetric, e.g.
Ax = −
B0
r sin ϑ
2
Ay = +
B0
r cos ϑ
2
(3.20)
while resorting to result (A.13) too we observe that in our p-representation
Lz = xpy − ypx = i~(py ∂px − px ∂py ) = −i~∂ϑ .
(3.21)
Thanks to (B.11) we then have
[HD , Jz ] = eσx [Ax , Lz ] + eσy [Ay , Lz ] + i~e (σ × A)z
~eB0 ~eB0 σx [−r sin ϑ, ∂ϑ ] + σy [r cos ϑ, ∂ϑ ] + i
σx r cos ϑ + σy r sin ϑ
= −i
2
2
=0
(3.22)
and conservation of Jz will consequently hold. It is then useful to separate the “radial” and
“angular” variables as
ψ1,m = u1,m (r)eimϑ
(3.23a)
while a glance at (3.11) and (3.26) as well as considering (B.17) should convince one that
ψ2,m = u2,m (r)ei(m+1)ϑ
(3.23b)
correctly pairs with (3.23a); m is the orbital angular momentum quantum number (see B.13),
but anyhow it is not a constant of the motion as the total angular momentum j is.
34
3.2
Solution within ordinary quantum mechanics
The aim of this section is to solve (3.4) in the standard context where the commutation relations
(2.14) and (2.15) hold, presenting an approach which will also be followed to face the more
complicated GUP equations. What we expect is to reobtain the well-known energy spectrum for
a relativistic electron first found out by Rabi in [20] and discussed for example in [4], given by
E=
p
M 2 c4 + 2~eB0 c(n + 1)
n = 0, 1, 2, ...
(3.24a)
p
M 2 c4 + 2~eB0 cn
n = 0, 1, 2, ...
(3.24b)
for the upper component and
E=
for the lower one. One deduces from this that only for ψ2 a state can exist where E coincides
with the rest energy, e.g.

ψ (0) = 
0
(0)
ψ2


(3.25)
and the two components ψ1 and ψ2 are then said to be “supersymmetric partners”.
We begin by going back to (3.8) and utilizing the standard representation (2.6): with the aid of
(3.19) and (3.20 the operators are now
eB0
e
(x + iy) = p+ − λ ∂px + i∂py
P+ = px + ipy + (Ax + iAy ) = p+ + i
c
2c
iλ
iϑ
=e
r − λ∂r − ∂ϑ
r
eB0
e
(x − iy) = p− + λ ∂px − i∂py
P− = px − ipy + (Ax − iAy ) = p− − i
c
2c
iλ
= e−iϑ r + λ∂r − ∂ϑ
r
(3.26)
(3.27)
as it is most quickly seen resorting to Eqn. (A.5) to (A.12) and having defined the new constant
λ=
~eB0
.
2c
(3.28)
Armed with these expressions, we are ready to make (3.14) and (3.15) explicit. One has
λ2
∂r −
r
λ2
P− P+ = r 2 − 2iλ∂ϑ + 2λ − λ2 ∂r2 − ∂r −
r
P+ P− = r 2 − 2iλ∂ϑ − 2λ − λ2 ∂r2 −
λ2 2
∂
r2 ϑ
λ2 2
∂
r2 ϑ
(3.29)
(3.30)
and applying the separation operated in (3.23) we notice that the problem lends itself to be
approached in a way formally equivalent to that used for a circular oscillator in two dimensions,
solved in polar coordinates. Following, for example, the treatment found in [7] and leaving
3.2. SOLUTION WITHIN ORDINARY QUANTUM MECHANICS
35
calculation details aside, we easily get the eigenvalues and eigenfunctions relative to (3.30)
r2
r2
(n )
eimϑ
ψ1,mr (r) = C1 r |m| e− 2λ 1 F 1 −nr , |m| + 1;
λ
h
i
ǫ2 = 2λ 2 (nr + 1) + m + |m|
(3.31)
(3.32)
and to (3.29)
r2
= C2 r
e
F
−n
,
|m
+
1|
+
1;
1
r
1
λ
h
i
2
ǫ = 2λ 2nr + (m + 1) + |m + 1|
2
|m+1| − r2λ
(n )
ψ2,mr (r)
ei(m+1)ϑ
(3.33)
(3.34)
where C1 , C2 are constants to be fixed by normalization, nr = 0, 1, 2, ... is the radial quantum
number and 1 F 1 is the hypergeometric confluent series (see Appendix C), which in these cases
reduces to a polynomial of degree nr so that the solutions won’t diverge for large values of r. It
is easily seen that the eigensolutions (3.31) and (3.33) are always normalizable; note also that
replacing m with m + 1 renders ψ1 exactly equal to ψ2 .
From (3.32) and (3.34) if we respectively define
n = nr +
m + |m|
= nr + s
2
n′ = nr +
m + 1 + |m + 1|
= nr + t
2
s=

0
m≤0
m m ≥ 1

0
m ≤ −1
t=
m + 1 m ≥ 0
(3.35)
(3.36)
we can observe that the energy levels are degenerate and they are given by (only positive roots
are physical meaningful here, as pointed out in Chapter 1)
E=
p
M 2 c4 + 2~eB0 c (n + 1)
p
M 2 c4 + 2~eB0 cn′
n = 0, 1, 2, ...
(3.37)
n′ = 0, 1, 2, ...
(3.38)
for the first component (spin up) and
E=
for the spin down one, so actually we have exactly found what was expected from the beginning
of the section. We also learn from (3.35) and (3.36) that only negative m integers couple (for ψ2 )
with ground state energy, and zero or negative ones do the same (for ψ1 ) with the first excited
energy; this is not something to worry about since reversing the magnetic field adjusts the
situation and globally all values of the orbital angular momentum quantum number will admit
states with “zero energy”, in one configuration or the other. However, we note in passing that this
is related to a remarkable feature of two-dimensional Dirac massive fermions which goes under
the name of parity anomaly (reversing the magnetic field just realizes a parity transformation),
pointed out years ago in [10].
Of course, states of the system are described by the two-components eigenstates of (3.7), so
we must put together the obtained results to find them. The lowest energy state is infinitely
36
degenerate with respect to all non-positive values of the orbital angular momentum




0
0
=
 eimϑ
ψ (0) = 
(0)
(0)
ψ2,m−1
u2,m−1
m = 0, −1, −2, . . .
E = M c2
(3.39)
while for the excited levels

 

(n −1)
(nr −1)
ψ1,mr
u1,m
=
 eimϑ
=
(nr )
(nr )
ψ2,m−1
u1,m
ψ (nr )
E=
nr = 1, 2, . . .
p
M 2 c4 + 2~eB0 cn
m∈Z
(3.40a)
(3.40b)
n = 1, 2, . . .
and it can be seen that also all the other energies have an infinite degeneracy for what concerns
negative m; regarding zero or positive values, instead, we observe that each level is (nr + t)-fold
degenerate. Note also that we may more correctly write (3.39) and(3.40a) in terms of the true
conserved number j by use of (B.17):
(0)
ψj
(nr )
ψj

0

0

1
≡  (0)  =  (0)  ei(j+ 2 )ϑ
ψ2,j− 1
u2,j− 1
2
2
 


(nr −1) i(j− 1 )ϑ
(nr −1)
2

ψ1,j− 21  u1,j− 21 e
≡
 =  (n )

(nr )
r
i(j+ 12 )ϑ
ψ2,j− 1
u2,j− 1 e
2
3.3

1 3
j = − ,− ,...
2 2
(3.41)
∀j .
(3.42)
2
Solution within a minimal length GUP
It is now time to put our Dirac equation (3.2) in the context of minimal length theories. We
achieve this first of all by recalling from Chapter 2, Eqn. (2.79) and (2.80), the aspect taken on
by position and momentum operators in the “modified” p-space. For convenience, here they are
again:
x̂i ϕ(p) = i~(1 + ηp2 )∂pi ϕ(p)
(3.43)
p̂i ϕ(p) = pi ϕ(p) .
(3.44)
The fundamental operators of our problem, P+ and P− , are then straightforwardly generalized
to the GUP case (and written in momentum polar coordinates, consult Appendix A for details)
3.3. SOLUTION WITHIN A MINIMAL LENGTH GUP
37
as
e
eB0
P+ = px + ipy + (Ax + iAy ) = p+ + i
(x + iy) = p+ − λ 1 + ηp2 )(∂px + i∂py
c
2c
iλ
(3.45)
= eiϑ r − 1 + ηr 2 λ∂r + ∂ϑ
r
eB0
e
(x − iy) = p− + λ(1 + ηp2 ) ∂px − i∂py
P− = px − ipy + (Ax − iAy ) = p− − i
c
2c
iλ
= e−iϑ r + 1 + ηr 2 λ∂r − ∂ϑ
.
(3.46)
r
and we also need to compute
P+ P− = r 2 + 2 1 + ηr 2
P− P+ = r 2 + 2 1 + ηr 2
−λ (1 + i∂ϑ ) − ηλ2 (r∂r − i∂ϑ ) +
λ (1 − i∂ϑ ) − ηλ2 (r∂r + i∂ϑ ) +
2
−λ
2
−λ
1 + ηr
1 + ηr
2 2
2 2
∂r2
1
1
+ ∂r + 2 ∂ϑ2
r
r
(3.47)
∂r2
1
1
+ ∂r + 2 ∂ϑ2
r
r
(3.48)
in order to write and solve the second order “squared” equations (3.14) and (3.15) which, taking
into account again the separation of variables operated in (3.23), read
d
2
2
−m +
r + 2λ 1 + ηr
m + 1 − ηλ r
dr
2
1 d
m2
2
2 2 d
−λ 1 + ηr
+
− 2
u1,m (r) = ǫ2 u1,m (r) (3.49)
dr 2 r dr
r
d
2
2
r + 2λ 1 + ηr
m − ηλ r
+m+1 +
dr
2
1 d
(m + 1)2
2
2 2 d
+
−
u2,m (r) = ǫ2 u2,m (r) . (3.50)
−λ 1 + ηr
dr 2 r dr
r2
Instead of suddenly committing ourselves to the study of these equations, we may “manually”
build-up zero energy solutions and discover under which circumstances they are normalizable,
thus gaining some insight into the nature the problem we are dealing with.
We go for P− ψ2 = 0 [ see (3.10) and (3.15], e.g.
d
(0)
λ(m + 1)
2
r + λ 1 + ηr
+
1 + ηr
u2,m (r) = 0
dr
r
2
yielding
(0)
u2,m = r −(m+1) 1 + ηr 2
1
− 2λη
(3.51)
(3.52)
38
of which we can study the normalizability; we recall here that the framework of our GUP model
renders it necessary to use a specific measure of integration, showed in (2.40). The quantity1
Z (0) 2
u2,m is finite for
d2 p
=
1 + ηp2
Z
r −2m−2 1 + ηr 2
−
1
λη
rdrdϑ
1 + ηr 2
(3.53)
1
.
λη
m < 0 ∧ m > −1 −
(3.54)
Much in the same fashion, requiring P+ ψ1 = 0 brings along
(0)
d
λm
2
+
1 + ηr
u1,m = 0
r − λ 1 + ηr
dr
r
2
solved to give
(0)
u1,m = r m 1 + ηr 2
and for it to be integrable
Z (0) 2
u1,m d2 p
=
1 + ηp2
Z
(3.55)
1
2λη
r 2m 1 + ηr 2
it would be necessary to satisfy
m > −1 ∧ m < −
(3.56)
1
λη
rdrdϑ
1 + ηr 2
(3.57)
1
λη
(3.58)
but this is never possible in our current assumption of a magnetic field directed along the
positive-z. Summarizing, we found


ψ (0) = 

0
r −(m+1)
(1 + ηr 2 )
1
2λη



ei(m+1)ϑ 
m = −1, −2, −3, ...
m>−
1
−1
λη
(3.59)
and a major difference stands out manifestly in comparing with the results of the previous section: not all the (negative) values of the orbital angular momentum quantum number m do
associate to a ground-state, but there’s a lower bound descending from the exponential not figuring anymore in the eigenfunctions. Thus the correct behaviour at infinity is not granted but
we had to impose it. Anyway, normalization alone is not enough to assure physical soundness of
a state, as we will discuss in a while.
With knowledge of the lowest energy solutions, we delve into the detailed study of (3.49) and
(3.50). They actually appear in the form
d2
d
−f (r) 2 + g(r) + hi (r) ui,m (r) = ǫ2 ui,m (r)
dr
dr
i = 1, 2
(3.60)
1
One has to remember that the Jacobian of the transformation to polar coordinates brings an additional r.
Details in appendix A.
39
where
f (r) = λ2 1 + ηr 2
2
(3.61)
2
λ2
1 + ηr 2
g(r) = −2ηλ2 1 + ηr 2 r −
r
λ2 m 2
2
h1 (r) = r 2 + 2λ(m + 1) 1 + ηr 2 + 2ηλ2 m 1 + ηr 2 +
1 + ηr 2
2
r
2 (m + 1)2
2
λ
1 + ηr 2 .
h2 (r) = r 2 + 2λm 1 + ηr 2 − 2ηλ2 (m + 1) 1 + ηr 2 +
2
r
(3.62)
(3.63)
(3.64)
We may now seek a transformation capable of leading us to equations of the Schrödinger kind,
hopefully easier to handle and to compare with those of some other known problem; this is
achieved by introducing
ui,m (r) = ρ(r)ϕi,m (r)
Z
ρ(r) = exp
χ(r)dr
Z
1
p
q=
dr
f (r)
1 d
d
=√
dr
f dq
(3.65)
(3.66)
(3.67)
(3.68)
from which follows
1
1
√
√
q = √ arctan( ηr) ⇐⇒ r = √ tan (qλ η)
λ η
η
π
q ∈ 0, √
2λ η
(3.69)
(3.70)
and (3.60) goes into (the primes indicate derivatives in r)
d2
− 2+
dq
2g − 4χf + f ′
√
2 f
d
2
′
+ hi + gχ − f χ + χ ϕi = ǫ2 ϕi .
dq
(3.71)
Our choice for χ(r) will then be
χ(r) =
and we also derive
1
2g + f ′
=−
4f
2r
R
ρ(r) = e
χ(r)dr
1
= r− 2
(3.72)
(3.73)
for later use; by means of such a transformation we have cast the equations into the form
d2
− 2 + Ui (q) ϕi (q) = ǫ2 ϕi (q)
dq
(3.74)
where the “potentials” are given by
Ui = hi + gχ − f χ2 + χ′ .
(3.75)
40
Calculating them explicitly them is a somewhat lengthy but otherwise easy operation providing
the expressions

1
m2 −

1
4
U1 (q) = − + ηλ2 
 sin2 (qλ√η) +
η

1
1
1
3
m+ +
m+ +
2 ηλ
2 ηλ 

√

cos2 (qλ η)

1
3
1
1
1
1
m+ +
m+
m− +
 m+ 2
1
2
2 ηλ
2 ηλ 
 .
+
U2 (q) = − + ηλ2 
√
√
2


2
η
cos ( ηλq)
sin (qλ η)

We define the shifted potential:
e (q) = U (q) + 1
U
η
and the shifted eigenvalues:
ǫ̃2 = ǫ2 +
(3.76)
(3.77)
(3.78)
1
η
(3.79)
so that the q-problems become:
d2
e
− 2 + Ui (q) ϕi (q) = ǫ̃2 ϕ1,2 (q)
dq
(3.80)
In this form, the problem can be compared with the one originating from the nonrelativistic
Pöschl-Teller potential hole for a particle of mass µ, a detailed study of which can be found in
[7]
ξ(ξ − 1)
V0 ζ(ζ − 1)
+
VPöschl-Teller (q) =
2 sin2 (νq) cos2 (νq)
π
0<q<
ζ>1
ξ>1
2ν
V0 =
~2 ν 2
µ
(3.81)
(3.82)
and giving rise to the differential equation
d2
− 2 + UPT (q) ϕ(q) = k2 ϕ(q)
dq
UPT =
2µVPT
.
~2
(3.83)
Jumping to the results, the eigenvalues and eigenfunctions turn out to be given by
k2 = ν 2 (ζ + ξ + 2n)2
nr = 0, 1, 2, ...
1
ζ
ξ
2
(nr )
ϕ
(q) = C [sin (νq)] [cos (νq)] 2 F 1 −nr , ζ + ξ + nr , ζ + ; sin (νq)
2
(3.84)
(3.85)
with 2 F 1 being the hypergeometric function: in the present case it is just a n-order polynomial.
C is a normalization constant.
41
In order to use these results we identify in (3.49) and (3.50) respectively
k2 = ǫ̃2
√
ν=λ η
(3.86)
(3.87)
1
ζ1 (ζ1 − 1) = m2 −
4 3
1
ζ2 (ζ2 − 1) = m +
m+
2
2
3
1
ξ1 (ξ1 − 1) = m + +
m+
2 ηλ
1
1
m−
ξ2 (ξ2 − 1) = m + +
2 ηλ
(3.88)
(3.89)
1
1
+
2 ηλ
1
1
+
.
2 ηλ
(3.90)
(3.91)
Let us concentrate first on the “upper” component. Solving (3.88) and (3.90) for the parameters
one finds
1
±m
2


+m +
1
1
=
ξ1 = ± m + 1 +
−m −
2
ηλ
ζ1 =
(3.92)
3
2
1
2
+
−
1
λη
1
λη
(3.93)
where the signs are to be chosen in order to fulfil (3.82). An immediate consequence of this is
that right now we cannot put m = 0 (but we will later see that this is not the ultimate word on
the topic), causing ζ1 to become less than one; careful selection of the signs in (3.93) is crucial
as well. Because of this we explicitly write the inequalities which have to be verified, namely

3


 +m + +
2
ξ1 =
1


 −m − −
2
1
1
1
> 1 =⇒ m > − −
λη
2 λη
1
3
1
> 1 =⇒ m < − −
.
λη
2 λη
(3.94a)
(3.94b)
Now let m be positive (m=1,2,...): in (3.92) the plus sign is to be selected, and for the other
parameter (3.94a) alone is satisfactory. On the other hand, negative values of m require the
minus sign in (3.92) but both choices are possible for ξ1 . More precisely, exploiting also relations
(3.84),(3.86)-(3.87) and (3.79), one has





ζ1 = 21 − m





1
1


1

m>− −
ξ1 = m + 32 + λη



2
λη





ǫ2 = 4λ(n + 1) [1 + λη (n + 1)]

r
r

m ≤ −1

1




ζ1 = 2 − m




1
3

1

m<− −
ξ1 = −m − 12 − λη



2
λη





ǫ2 = 4λ(n − m) [λη (n − m) − 1]
r
r
(3.95a)
(3.95b)
42
m≥1



ζ =

 1
1
2
+m
1
ξ1 = m + 32 + λη



ǫ2 = 4λ(n + m + 1) [1 + λη (n + m + 1)]
r
r
(3.96)
We then make the same analysis for ψ2 . From (3.89) and (3.91) we read

+m + 3
1
2
ζ2 = ± (m + 1) =
−m − 1
2
2

1
+m+
1
1
= 2
ξ2 = ± m +
1 − m −
2
ηλ
2
(3.97)
1
λη
1
λη
(3.98)
Here bearing in mind condition (3.82) results in m = −1 not being a valid parameter. Once
again, we guess this could be a shortcoming of our current treatment, but we pass on for now.
To clarify things we distinguish the range of validity of each possibility for ξ2 :

1


 +m+
2
ξ2 =
1


 −m−
2
1
1
1
> 1 =⇒ m > −
λη
2 λη
1
1
1
> 1 =⇒ m < − −
.
λη
2 λη
(3.99a)
(3.99b)
The spectra descending from the various options are again calculated with ease





ζ2 = −m − 21





1
1


1

m> −
ξ2 = 12 + m + λη



2
λη





ǫ2 = 4λn [1 + ληn ]

r
r

m ≤ −2




ζ2 = −m − 12






1
1

1

m<− −
ξ2 = 21 − m − λη



2
λη





ǫ2 = 4λ(n − m) [λη (n − m) − 1]
r
r



ζ = m + 23

 2
∧
1
ξ2 = 12 + m + λη


1
1 
ǫ2 = 4λ (n + m + 1) [1 + λη (n + m + 1)]
m> −
r
r
2 λη
(3.100a)
(3.100b)
m≥0
(3.101)
We may now pause and examine the outcomes of our dissection. Perhaps the first thing to note
is that in the limit η → 0 the spectra found in (3.95a),(3.96),(3.100a) and (3.101) correctly
go to the corresponding no-GUP ones found in the previous section; the other options (3.95b)
and (3.100b) are no longer available and the situation is simplified as it should. However, not
everything is fine: in fact the values m = 0 and m = −1 for the angular momentum quantum
number have been left out, and they are not recovered in the aforementioned limit, apparently
preventing us to retrieve the complete and exact set of ordinary-QM solutions which does not
exclude any m. More comprehensively, in switching to the Pöschl-Teller potential (3.76) for ψ1
43
we have been forced due to (3.82) to assume [see (3.92) and (3.94]
m 6= 0
1
1
1
3
,− −
m∈
/ − −
2 λη
2 λη
∧
(3.102)
and analogously the ψ2 equation (3.77) has required for its use [from (3.97), (3.99)]
m 6= −1
1
1
1
1
,+ −
m∈
/ − −
2 λη
2 λη
∧
(3.103)
so we are excluding some quantum numbers, and attention must be paid in order to verify
whether the above restrictions are really physical ones or, more sensibly, only mathematical
issues. In this respect, note also that relations on the right are of no importance when η → 0,
while those on the left retain their effects. For a better comprehension, it is useful to restate our
solutions in the space of the variable q in terms of the “radial momentum” r. From (3.69) follows
elementarily
√
sin(νq) = sin(qλ η) = p
√
cos(νq) = cos(qλ η) = p
√
ηr
1 + ηr 2
1
1 + ηr 2
(3.104)
(3.105)
and by means of (3.65), (3.73) and (3.85) we can write the radial eigenfunctions of our problem
as
(n )
ui,mr (r)
"
#ζi "
#ξi
ηr 2
1
= Ai r
2 F 1 −nr , ζi + ξi + nr , ζi + ;
2 1 + ηr 2
1 + ηr 2
1 + ηr 2
1
r ζi − 2
ηr 2
1
= Ci
F 1 −nr , ζi + ξi + nr , ζi + ;
(3.106)
ζi +ξi 2
2 1 + ηr 2
(1 + ηr 2 ) 2
− 21
p
√
ηr
p
1
and then
(n )
(n )
(3.107)
(n )
(n )
(3.108)
r
ψ1,mr (r, ϑ) = u1,m
(r)eimϑ
r
ψ2,mr (r, ϑ) = u2,m
(r)ei(m+1)ϑ .
What we shall now do is studying the convergence of the normalization integrals for these solutions and, more generally, finding out which are the conditions for them to be sensed of physical
grounds, thus investigating whether (3.102)-(3.103) are essential and inescapable. Actually a
closer look at the subject does not only suggest us to require normalizability of the eigensolutions but also their having a finite uncertainty in momentum (or equivalently their being in the
domain of p̂2 , since (∆p)2 = hp2 i − hpi2 . See Chapter 2 and reference [11]). We may then study
E Z Z rdrdϑ D
(nr ) 2
(nr ) (nr )
ψ
ψi,m ψi,m =
1 + ηr 2 i
2
Z ∞
rdr
r 2ζi −1
ηr 2
1
2
= 2π |Ci |
F 1 −nr , ζi + ξi + nr , ζi + ;
1 + ηr 2 (1 + ηr 2 )ζi +ξi 2
2 1 + ηr 2
0
44
that is
D
(n ) (n )
ψi r ψi r
E
2
= 2π |C|
Z
∞
I(r)dr
(3.109)
0
with I(r) having the asymptotic behaviour
r→0
(3.110)
r→∞
(3.111)
I(r) ∼ r 2ζi
I(r) ∼ r −2ζi −2
and together
D
(n )
(n ) ψi,mr p2 ψi,mr
E
2
= 2π |C|
Z
∞
r 2 I(r)dr
(3.112)
0
has to converge. This is furthermore related to the mean value of the squared relativistic energy
E 2 = m2 c4 + p2 c2 having to be finite. Thus putting all together the comprehensive necessary
and sufficient conditions for the physical existence of each state are
1
2
(3.113a)
1
.
2
(3.113b)
2ζi > −1 =⇒ ζi > −
−2ξi < −1 =⇒ ξi >
These, and not the ones descending from the Pöschl-Teller potential expressions, ought to be
the real delimiters of the physically allowed zone. This pushes ourselves to review and extend
the validity of the results found above. Focusing on the upper component, (3.95) and (3.96) can
be merged into




(nr )


u1,m



1
m > −1 −
λη 

ǫ2





s




 (nr )
1 u1,m
m < −1 −
λη 



ǫ2
= C1
r |m| 2 F 1 −nr , nr + 2s + 2 +
1
λη , |m| +
2
ηr
1; 1+ηr
2
1
(1 + ηr 2 )s+1+ 2λη
(3.114)
h
i
= 4λ (nr + s + 1) 1 + λη (nr + s + 1)
=
m+|m|
2
= C1
r |m| 2 F 1 −nr , nr + 2 |m| −
1
λη , |m| +
1
(1 + ηr 2 )|m|− 2λη
h
i
= 4λ (nr + |m|) λη (nr + |m|) − 1
2
ηr
1; 1+ηr
2
(3.115)
45
while an analogue reasoning for the lower wavefunction leads to




(nr )


u2,m



1
m>−
λη 

ǫ2





t




 (nr )
1 u2,m
m<−
λη 



 ǫ2
= C2
r |m+1| 2 F 1 −nr , nr + 2t +
1
λη , |m
2
ηr
+ 1| + 1; 1+ηr
2
1
(1 + ηr 2 )t+ 2λη
h
i
= 4λ (nr + t) 1 + λη (nr + t)
=
m+1+|m+1|
2
= C2
r |m|−1 2 F 1 −nr , nr + 2 |m| −
1
ηr 2
1
λη , |m| ; 1+ηr 2
(1 + ηr 2 )|m|− 2λη
h
i
= 4λ (nr + |m|) λη (nr + |m|) − 1
(3.116)
(3.117)
These are the eigenfunctions and the associated eigenvalues solving our problem in the context
of a minimal length GUP theory. As seen from direct calculation, zero energy states can only
have a lower nonzero component and putting nr = 0 for the negative m in (3.116) immediately
results in (3.59). In the η → 0 limit the well-know situation of ordinary quantum mechanics is
recovered, since only (3.114) and (3.116) remain, with their associated spectra smoothly joining
(3.32)-(3.34), and also it is not difficult to verify that
1
ηr 2
r2
lim 2 F 1 −nr , a +
, b;
= 1 F 1 −nr , b;
η→0
ηλ
1 + ηr 2
λ
(3.118)
so that the ordinary eigenfuctions correctly reappear in this limiting case.
In what remains we concentrate only on solutions (3.114) and (3.116), which appear the natural
generalization of the ordinary quantum mechanical results, to which they return if we “turn off”
the GUP perturbation by letting η → 0. The cases (3.115)-(3.117) appear more delicate and
perhaps a bit tricky, first of all because they do not have any corresponding eigenfunction nor
energy spectra in the null-η standard context, as previously mentioned; we may wish to leave the
check of their being really legit to further investigations on the subject. However, on a physical
standpoint we can observe the following: the present problem is in many respect formally similar
to that involving a two-dimensional isotropic oscillator with the characteristic Landau frequency
eB0
Mc
ωL =
(3.119)
and whose solutions are, in the quantum relativistic case (see [25]) as well as in the well-know
Schrödinger theory, of the form
− 12
ψ(x) ∝ e
|x|
lc
2
(3.120)
where lc is the characteristic length of the oscillator, giving the spatial scale where the particle
is essentially confined, and reads
lc =
r
~
M ωL
(3.121)
46
in the present case of an “oscillating” electron. Now
√ √ 2
~ η eB0 M
~eB0
1 ~ η 2 1 ∆x0 2
λη =
=
η=
=
2c
2~M c
2
lc
2
lc
(3.122)
and since we just cannot probe the physics of an oscillator whose characteristic length is of the
order of the minimal uncertainty ∆x0 or below, we are led to require
1
≫1
λη
λη ≪ 1
(3.123)
telling us that (3.115) and (3.117) are not cases of direct interest, at least in a first analysis.
States of the electronic system will then be, adopting the same notation (3.35):

ψ (0) = 
0
(0)
ψ2,m−1


=
0
(0)
u2,m−1
m = 0, −1, −2, . . .

 eimϑ
m−1 > −
E = M c2
(3.124)
1
λη
and the excited levels
ψ (nr )
E=
s

 

(n −1)
(nr −1)
ψ1,mr
u1,m
=
 eimϑ
=
(nr )
(nr )
ψ2,m−1
u2,m−1
~eB0
n
M 2 c4 + 2~eB0 cn 1 + η
2c
nr = 1, 2, . . .
=
s
M 2 c4
m−1>−
+
1
λη
2M c2 ~ω
(3.125a)
M ~ωL
n
Ln 1 + η
2
(3.125b)
n = 1, 2, . . .
with the modified spectrum made of (3.124) and (3.125b) plotted in Figure 3.1 together with
the ordinary-QM one of Eq.(3.40b). As said for the no-GUP eigensolutions, one can better use
the true conserved quantum number j and rewrite (3.124) and (3.125a) as
(0)
ψj
(nr )
ψj

0


0

1
≡  (0)  =  (0)  ei(j+ 2 )ϑ
ψ2,j− 1
u2,j− 1
2
2
 


(nr −1) i(j− 1 )ϑ
(nr −1)
2
u
e
ψ

 1,j− 12   1,j− 21
≡
 =  (n )

1
(nr )
r
i(j+ 2 )ϑ
ψ2,j− 1
u2,j−
1e
2
1 3
j = − ,− ,...
2 2
j−
1
1
>−
.
2
λη
j−
1
1
>−
2
λη
(3.126)
(3.127)
2
If one wishes to compare the obtained results with those in [15] where the very same problem
is discussed, it has to be pointed out that there the authors chose a different and simplified
two-dimensional generalized commutation relation between the canonical operators, expectedly
leading to a different spectrum. In fact, it can be seen that the last term under the square root
in (3.125b) differs for a factor
1
2
from the corresponding one in the cited reference. Comparison
3.4. MASSLESS CASE AND APPLICATION TO GRAPHENE
Figure 3.1: Plot of ǫ2 vs n.
Blue: Ordinary QM spectrum with η = 0, λ =
Red : GUP spectrum with η = λ = 41
47
1
4
with the treatment given in [16] is instead more instructive, as the author adopts an approach
which is strictly analogue to our one in analysing the nonrelativistic limit of the Dirac Equation
(Pauli Hamiltonian) in presence of a magnetic field. In this limit our spectrum (3.125b) goes
E ≃ M c2 + ~ωL n +
ηM
(~ωL n)2
2
(3.128)
which can be shown to be in agreement (at first order in η) with the spin-dependent expression
found in the mentioned work.
3.4
Massless case and application to graphene
In certain contexts such as those emerging from the study of graphene layers and their under many aspects exceptional characteristics, electrons can be perfectly modelled as massless
fermions.
Graphene is the name given to a sheet of carbon atoms (a single graphite sheet) arranged to
form an honeycomb lattice, with strong σ-bonds due to sp2 hybridization of carbon orbitals
giving high in-plane cohesion while the remaining valence electrons (one for each atom) belong
to π orbitals which are formed by hybridizing two overlapping parallel p orbitals; these electrons are substantially delocalized, and explain the rather good conductivity observed along
the planes. It is an extraordinary material since with its discovery we have at our disposal a
real bi-dimensional system in which many exact models of quantum-electrodynamics (QED, of
which the Dirac equation is the cornerstone) can be tested due to the electrons’ neat relativistic
behaviour, which thus cease to be described by the Schrödinger equation and should manifest
all the features theorized by Dirac’s one and in particular those of massless relativistic particles,
48
(a) Honeycomb lattice and primitive cell.
(b) Graphene layer.
Figure 3.2: Graphene.
but with a speed which is considerably lower than that of light,thus hopefully making it possible
to experimentally detect and verify many theoretical predictions which have been out of direct
measurement reach for a long time.
In fact from an accurate solid state physics analysis emerges that at certain points of graphene’s
reciprocal lattice (the so-called “Dirac points”) the band structure provides two intersecting cones
and the dispersion relation for electrons becomes linear, indicating they behave like relativistic
massless particles: this means that the Dirac Hamiltonian for these low-energy electrons will
read
H D = vF α · p
where vF ≈
1
300 c
(3.129)
is the Fermi velocity. This in brief is one of the aspects that makes this newly
discovered (but simple) material a very promising one for technological applications, due to its
great transport properties and high free-electron speeds resulting from the particular periodicity
of the potential arising from the honeycomb arrangement.
The analysis carried out in this chapter may then be usefully repeated in the present case without
much effort, noticing that the current situation is reproduced if we put M = 0 and ǫ2 =
Then the ordinary quantum mechanical spectrum (3.40b) becomes
E = vF
r
2~eB0
n
c
E2
2 .
vF
(3.130)
and, most interestingly, the introduction of the GUP is observed to affect the dynamics of
massless fermions (sometimes called Dirac fermions) too, in such a way that
E = vF
s
2~eB0
~eB0
n 1+η
n
c
2c
and this is certainly something we would like to underline.
(3.131)
Conclusions
Many problems belonging to several branches of quantum mechanics have been lately placed
in the domain of generalized uncertainty principle theories and light has been shed on various
traits appearing in this context. The subject is finding a steady if not raising interest for various reasons, chiefly because models of this kind seem to be implicated by and interlaced with
quantum gravity, string theories, black holes cosmology as well as noncommutative geometries,
and could be seriously involved in a future widening of physical landscapes. The study of the
present problem has seen ourselves aiming to give another little contribute in this direction,
concentrating on a problem whose attentive consideration was still lacking in the literature.
After introducing the Dirac equation along with some of its general features, and having presented the GUP models with detailed consideration of those who only provide a minimal nonzero
uncertainty in position measurements, in the last chapter the goal of this entire work was
achieved, namely the exact solving of the (2+1) Dirac equation for massive electrons subjected
to an uniform static magnetic field, first in the ordinary quantum mechanical context and then
inside a minimal length GUP theory. Energy spectra as well as the corresponding eigenfunctions
for the two cases have been calculated and may be compared: the appearance of a linear term
in η under square root in the energy eigenvalues results as the effect of the minimal length
assumption. In the end we have extended the results to massless fermions having a linear dispersion relation, which is something concretely happening for graphene electrons: suitable measures
made on such systems could possibly experimentally reveal some theorized features as well as
help estimating the magnitude of the minimal length parameter η.
Surely the results we have found out demand a deeper and more detailed inspection of some
more subtle mathematical aspects as well as physical ones we have not delved into; they also
open the way for the computation of certain significant quantities for the system (Hall conductivity, Zitterbewegung to name two) which could help one to more fully appreciate the elements
of novelty brought in by the introduction of a minimal measurable length. Other possible calculations could regard the projection of the momentum space eigensolutions onto maximally
localized states, finding the quasiposition representation for the electrons bound states.
49
Appendix A
Expressions in polar coordinates
NOTE: To use the equations of this appendix in Chapter 3, one has to operate the replacements
x → px , y → py , Θ ± → p± .
The polar reference frame is a two-dimensional system of coordinates (r, θ) related to the Cartesian one through the transformation
Figure A.1: Polar reference frame in the xy plane.

p
r = x2 + y 2
ϑ = arctan y (A.1)
x
which is obviously invertible, giving

x = r cos ϑ
y = r sin ϑ .
(A.2)
The theory of multiple integrals states that if one needs to change variables of integration through
a certain transformation, the Jacobian of the latter must appear in the integral. In this case
∂x
∂(x, y) ∂r
∂(r, ϑ) = ∂y
∂r
cos ϑ −r sin ϑ
=
∂y sin ϑ r cos ϑ
∂ϑ
∂x
∂ϑ
50
=r
(A.3)
51
and it follows
ZZ
f (x, y)dxdy =
ZZ
f (r, ϑ)rdrdϑ .
(A.4)
Now let us introduce the variables Θ± = x ± iy. They assume the polar form
Θ+ = r(cos ϑ + i sin ϑ) = reiϑ
(A.5)
Θ− = r(cos ϑ − i sin ϑ) = re−iϑ
(A.6)
Θ+ Θ− = Θ− Θ+ = r 2 .
(A.7)
and moreover
When handling the partial differential operators
∂
∂x
and
∂
∂y
one may wish to know how they
act when we switch to polar reference; this can be done using the standard transformation of
differential operators based on the chain rule
∂
∂r ∂
∂ϑ ∂
∂
sin ϑ ∂
=
+
= cos ϑ
−
∂x
∂x ∂r ∂x ∂ϑ
∂r
r ∂ϑ
∂r ∂
∂ϑ ∂
∂
cos ϑ ∂
∂
=
+
= sin ϑ +
.
∂y
∂y ∂r
∂y ∂ϑ
∂r
r ∂ϑ
Now if we define
∂± =
∂
∂
±i
∂x
∂y
(A.8)
(A.9)
(A.10)
the use of (A.8) and (A.9) leads to
cos ϑ + i sin ϑ ∂
∂
i
+i
= eiϑ ∂r + ∂ϑ
∂r
r
∂ϑ
r
∂
cosϑ − i sin ϑ ∂
i
−iϑ
∂− = (cos ϑ − i sin ϑ)
−i
=e
∂r − ∂ϑ
∂r
r
∂ϑ
r
∂+ = (cos ϑ + i sin ϑ)
(A.11)
(A.12)
and other operators for which we can usefully get polar expressions are
x∂y − y∂x = r cos ϑ sin ϑ∂r + cos2 ϑ∂ϑ − r sin ϑ cos ϑ∂r + sin2 ϑ∂ϑ = ∂ϑ
1
1
∂+ ∂− = ∂− ∂+ = ∂x2 + ∂y2 = ∂r2 + ∂r + 2 ∂r2 .
r
r
(A.13)
(A.14)
Appendix B
Angular momenta in a
(2+1)-dimensional Dirac theory
NOTE: all the results obtained in the following hold true even for a GUP theory, since the
explicit expressions of angular momentum operators are not changed by the latter.
From the Dirac Hamiltonian appearing in
i~
∂Ψ
= HD Ψ
∂t
(B.1)
one is able to study which quantities are conserved during the evolution of a certain system
described by the Dirac equation, extending the Heisenberg relation of nonrelativistic quantum
mechanics
dA
i
∂A
= − [H, A] +
dt
~
∂t
(B.2)
for the evolution of a certain observable in a system governed by the Hamiltonian H.
Here we focus on angular momenta in the special simplified case of a (2+1) Dirac equation
(Dirac matrices being 2 × 2, i.e. Pauli matrices: αi = σi , β = σz ), which is the one of Chapter 3.
For a free fermion of mass m
HD = cα · p + βmc2
(B.3)
the commutator with the only nonzero component of the orbital angular momentum gives
[HD , Lz ] = [cα · p + βmc2 , xpy − ypx ]
= cαx [px , x]py − cαy [py , y]px = −i~c(αx py − αy px ) = −i~c(α × p)z
(B.4)
where we have made use of the fact that Dirac matrices are only active on the spin space and
of the property [A, BC] = B[A, C] + [A, B]C. The orbital angular momentum alone then is not
conserved.
52
53
For the spin we have the operator Sz = 2~ σz and so, using [σi , σj ] = 2iǫijk σk
~
~c
[cα · p + βM c2 , σz ] = [αx px + αy py + βmc2 , σz ]
2
2
~c
([σx , σz ]px + [σy , sz ]py ) = i~c(σx py − σy px ) = i~c(α × p)z
=
2
[HD , Sz ] =
(B.5)
which permits us to state that not even the spin is conserved, but on the other hand the total
angular momentum Jz = Lz + Sz verifies
[HD , Jz ] = [HD , Lz ] + [HD , Sz ] = −i~c(α × p)z + i~c(α × p)z = 0
(B.6)
being the true conserved quantity.
In presence of a magnetic field (automatically directed along the direction z normal to the plane,
cause otherwise the particle would exit the plane itself), (B.3) now reading
e HD = cα · p + A + βmc2 = cα · p + βmc2 + eα · A
c
A = (Ax , Ay )
(B.7)
(B.8)
we will have
[HD , Lz ] = −i~c(α × p)z + e[α · A, Lz ] = −i~c(α × p)z + eσx [Ax , Lz ] + eσy [Ay , Lz ] .
(B.9)
Furthermore, for the spin now it is
e~ e~
Ax [σx , σz ] + Ay [σy , σz ]
[α · A, σz ] = +i~c(α × p)z +
2
2
= +i~c(α × p)z + i~e (σ × A)z
(B.10)
[HD , Sz ] = i~c(α × p)z +
and the total angular momentum commutator with the Hamiltonian yields
[HD , Jz ] = eσx [Ax , Lz ] + eσy [Ay , Lz ] + i~e (σ × A)z .
(B.11)
Only if this quantity equals zero Jz will be conserved even in presence of a magnetic field.
When the total angular momentum is a constant of the motion, we can associate to it a “good”
quantum number. Denoting it j with j =
n
2
, n ∈ Z, we expect
Jz Ψ = ~jΨ
(B.12)
54
APPENDIX B. ANGULAR MOMENTA IN A (2+1)-D DIRAC THEORY
to always hold true if it does so at a certain beginning time. On the other hand, the quantum
numbers of the orbital angular momentum (m ∈ Z) and spin (s = ±1), that is
Lz Ψ = ~mΨ
~
Sz Ψ = sΨ =⇒ σz Ψ = sΨ
2
(B.13)
(B.14)
(B.15)
are not fixed and can change during the course of time. However, by definition of Jz the relation
j =m+
s
2
(B.16)
is true at any time and can be restated by saying that, given a state which describes our system
at a certain instant,
m=

j −
j +
1
2
1
2
if s =
1
2
if s = − 12
having distinguished spin-up and spin-down states.
(B.17)
Appendix C
Hypergeometric functions
One of the most encountered equations in Physics is the so-called hypergeometric equation
z(1 − z)
dg(z)
d2 g(z)
+ [c − (a + b + 1)z]
− abg(z) = 0
2
dz
dz
(C.1)
where z is a complex variable and a, b and c are complex parameters. It is a differential equation
of the second order with three singular points of the Fuchsian kind in y = 0, 1, ∞; in other
words, the (generally two) linearly independent solutions g(z) of (C.1) can be expressed, in a
neighbourhood of the mentioned points of the complex plane, through Laurent series with only a
finite number of negative powers. Since the equation is symmetric with respect to the exchange
of the parameters a and b, this will also be a property of g(z): a and b play the very same role,
different from that of c.
The solution can be expressed by means of the Riemann p-symbol as
g(z) = P





0
1
0
0



 1−c c−a−b
∞
a
z
b





(C.2)




and this permits us to formally write down the solutions valid around the three Fuchsian singular
points in terms of functions which are analytic in those points (see [21] for details). Searching
for one of the two solutions (denoted g1 (z)) holding in a neighbourhood of z = 0 in the form of
a power series, e.g.
g1 (z) =
∞
X
dk z k
(C.3)
k=0
one can show that the coefficients are exactly determined to be
dk =
1 (a)k (b)k
k! (c)k
(C.4)
with the standard notation
(a)k = a(a + 1)(a + 2) . . . (a + k − 1) =
55
Γ(a + k)
Γ(a)
(a)0 = 1
(C.5)
56
APPENDIX C. HYPERGEOMETRIC FUNCTIONS
where Γ is Euler’s Gamma function. This solution is named hypergeometric function and is then
given by
g1 (z) ≡ 2 F 1 (a, b, c; z) =
∞
X
(a)k (b)k z k
(c)k
k=0
|z| < 1
k!
(C.6)
where the ray of convergence is determined by the first singularity reached in the complex plane.
By the way, the other solution valid around z = 0 is determined through the properties of the
Riemann P-symbol to be
g2 (z) = z 1−c 2 F 1 (a − c + 1, b − c + 1, 2 − c; z) .
(C.7)
From (C.4) it is straightforward to realize that for every null or positive integer n
(−n)k = (−n)(−n + 1) . . . (−n + k − 1) = (−1)k
(−n)k = 0 ∀k > n
n!
(n − k)!
0≤k≤n
(C.8)
(C.9)
and this implies that when either a = −n or b = −n respectively the hypergeometric functions
2 F 1 (−n, b, c; z)
or 2 F 1 (a, −n, c; z) are no else than polynomials of degree n, because
dk = 0
∀k > n
(C.10)
and the series (C.6) is truncated after n + 1 terms, explicitly becoming
n
n
X
X
(−n)k (b)k z k
k n (b)k k
=
z
(−1)
2 F 1 (−n, b, c; z) =
(c)k
k!
k (c)k
(C.11)
k=0
k=0
and similarly for the case b = −n. It is also useful to note that
2 F 1 (0, b, c; z)
= 2 F 1 (a, 0, c; z) = 1
2 F 1 (a, b, c; 0)
=1.
(C.12)
The name hypergeometric descends from observing that, since (1)n = n! , the relation
2 F 1 (a, 1, a; z)
= 2 F 1 (1, b, b; z) =
∞
X
zn =
k=0
1
1−z
|z| < 1
(C.13)
is true, thus showing that in this particular case the hypergeometric function (also called hypergeometric series) becomes the usual geometric series whose representation within the circle of
radius 1 is well-known.
Let us now consider again the starting equation (C.1). By operating the change of variable
z=
x
b
x = bz
(C.14)
the singularities are now shifted to the points x = 0, b, ∞: this is obviously only a formal
difference. However, we may check what happens if we move the second Fuchsian singularity to
57
∞, i.e. if we perform the limit operation b → ∞, resulting in the new equation
dg(x) a
d2 g(x) c
+
−
1
− g(x) = 0
dx2
x
dx
x
(C.15)
xg′′ (x) + (c − x)g ′ (x) − ag(x) = 0
(C.16)
otherwise stated as
which takes the name of confluent hypergeometric equation or Kummer equation. In fact, two
distinct (regular) singularities have been joint together into an only one at infinity, with the
consequence of the latter now representing an irregular singularity. It is not difficult to deduce
the solutions of (C.16) if we follow the same path which brought us to it: in terms of the variable
x, it must be
∞
x X (a)k (b)k z k
g1 (x) = 2 F 1 a, b, c;
=
b
(c)k bk k!
(C.17)
k=0
and in our limit
lim (b)k = bk
(C.18)
b→∞
so that eventually
g1 (x) ≡ 1 F 1 (a, c; x) =
∞
X
(a)k z k
k=0
(c)k k!
.
(C.19)
This special solution goes under the name of hypergeometric confluent function (or series) in
the variable x and with parameters a and c.
A noteworthy example may be
1 F 1 (a, a; x)
= ex
(C.20)
as it is immediate to prove; moreover, in the same way as it was for 2 F 1 , setting the first
parameter (but not the second, here) equal to a negative (or zero) integer one has
1 F 1 (−n, c; x)
=
n
X
(−n)k xk
k=0
(c)k k!
=
n
X
(−1)k n
k=0
(c)k
k
xk
(C.21)
Investigation of these polynomials reveals that they are proportional to the Laguerre ones.
Bibliography
[1] R. J. Adler, P. Chen, D. I. Santiago, The Generalized Uncertainty Principle and Black Hole
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Acknowledgements
I would like to heartily thank dr. Orlando Panella for having offered me to take part in the solution of a problem which had never been treated in the literature and to make it the subject of
my Bachelor’s thesis. His being willing and helpful towards me have been the best introduction
I could wish to scientific research.
I am grateful in the same way to prof. Pinaki Roy of the ISI of Calcutta for his constant availability to explain me all the details and to be at my side in overcoming all the difficulties I
and we have encountered in facing these topics. His kindness and his prompt answering when
corresponding with me have been really fundamental.
I would also thank all those professors who, during these three years, have incisively contributed
in my and my classmates’ training, thoroughly committing themselves to our improvement, often
going over what was simply due and showing their human side too. Even if I were to forget the
contents of their teaching, the quality of the latter will stand firm in my mind and heart.
My thoughts go then to those who, three years ago, were starting with me this adventure so
demanding and difficult at the beginning but which has turned into a truly satisfying experience,
and to all the others met along the way; to the more or less tight friendships born inside the
department’s walls, each of which has given me something I can now bring with me. Thanks for
our helping each other, for the troubles experienced together, for the shared hopes, disappointments and aspirations, the chats along the corridors or just outside the entrance, the afternoons
and mornings in the laboratories, what we’ve said and what we have not, what has joined and
what has divided us, but has helped us becoming a bit more true. Above all, thanks for teaching
me how it is sometimes good not to take oneself too seriously.
So many people have been part of my life in these three years and for sure in a few lines I cannot
recall all of them as they would deserve. But how could I leave without mentioning “the Trio”
and my faithful fellows Reddie and John, whose friendship is stronger than every storm, true
raisers of my spirits, without whom my life would not be the same at all. And then Fabio and
all the evenings spent together, the dinners, the games, which so big a role have played in easing
the weariness of the studies when it was most burdening.
Warmest thanks to all the Noé Cell, my journey with whom just began when I was getting
into University: you have accompanied and sustained me, being the safe harbour where I could
rest my sails whenever they wore out. Together with all boys and girls of the Parish Church of
Castel del Piano, you’ve showed me what the inner simplicity of the heart is; while on one hand
I have been finding out what marvellous immensity is here for us to discover in books and all the
creation, I’ve been gifted with another richness as well, the one coming from people around me
and which cannot be learnt, you just live it. Thanks also to the “Gemma” group, for the music
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we believe in and for what we’ve done and, even more, we will do. A special thank you goes to
Don Francesco, for his being a Faith witness and a bright and sound guide, able to get the best
out of me.
This work and the years of study I’ve done to get to this point would have never been possible
without the extraordinary support of all my family. Speaking about my gratitude towards each
of you, my dear mum and dad, my wonderful sisters Giulia and Costanza and all the grandparents (even who’s not here with us anymore) is far more than restrictive. You are those who,
more than anyone else, day by day teach me what words like care, devotion, love are about. I
am infinitely thankful for this. What you’ve all along done for me, to let me concentrate on my
studies and create the best possible conditions for me to succeed, cannot be wholly told and
there’s not so much more I could say. The only true manner for me to respond to such great gifts
that I am being given is to put all myself in what the Lord is calling me to do, and I hope I’ve
achieved this at least in part, but what I hope over all things is to make you aware (if you’re not
yet so) that every letter (and symbol!) written on this thesis and the little I’ve reached as far as
now, not only in my studies but in every other thing I’ve engaged myself into, is concretely and
for a great share a work of your doing. Thank you.
Last but certainly non least is my thanks to He who has wanted this and called it to existence.
Because I was not there to choose where, how and when to be born, and not even what my skills
would be. Thanks to Him who numbers my steps and leads them onto the paths of Light. To
Jesus, the Star whose Love glows in my life more than everything else.
Ringraziamenti
Ringrazio di cuore il dott. Orlando Panella per avermi fatto la stimolante proposta di prendere
parte alla risoluzione di un problema ancora non trattato in letteratura e di farne l’argomento
di questo lavoro di tesi triennale. La disponibilità e la fiducia mostrate nei miei confronti sono
state la migliore introduzione all’ambiente della ricerca scientifica.
Sono altrettanto grato al prof. Pinaki Roy dell’ISI di Calcutta per il suo continuo prestarsi a
spiegarmi tutti i dettagli e a superare insieme a me tutte le difficoltà che ho e che abbiamo
incontrato nell’affrontare questi argomenti. La sua cortesia e la sua prontezza nel corrispondere
con me sono state fondamentali.
Vorrei inoltre ringraziare tutti i professori che, in questi tre anni, hanno contribuito incisivamente
alla formazione mia e dei miei compagni di corso impegnandosi senza riserve per la crescita di noi
studenti, andando spesso oltre il semplice dovere e mostrando anche il loro lato umano. Anche
se ne dimenticassi le nozioni, gli insegnamenti di qualità restano scolpiti nella mente e nel cuore.
Il pensiero va poi a chi tre anni fa iniziava con me questa avventura all’inizio cosı̀ difficile e
dura ma rivelatasi enormemente appagante, e a tutti gli altri conosciuti lungo il cammino; alle
amicizie più o meno strette nate tra le mura del dipartimento, ciascuna a suo modo in grado di
donarmi qualcosa che ora porto con me. Grazie per l’aiuto che ci siamo dati, le difficoltà vissute
insieme, le speranze, le delusioni e le aspirazioni condivise, le chiacchierate lungo i corridoi o
davanti all’ingresso, i pomeriggi e le mattinate nei laboratori, le cose dette e anche quelle non
dette, ciò che ci ha unito e ciò che ci ha a volte diviso, ma ci ha aiutato a diventare un po’ più
veri. Grazie soprattutto per avermi insegnato che a volte è bene non prendersi troppo sul serio.
Tante persone sono state parte della mia vita in questi tre anni e ricordarle tutte non è certo cosa
che possa pensare di fare a dovere in poche righe. Non potrei però terminare senza menzionare
almeno il “Trio” e i fedeli compari Reddie e John, la cui amicizia è più forte di qualunque
tempesta, veri coltivatori del mio entusiasmo, senza i quali la mia vita non sarebbe la stessa.
E poi Fabio e tutte le serate trascorse insieme, le cene, i giochi, che tanta parte hanno avuto
nell’alleggerire i carichi dello studio quando essi si facevano più sentire.
Un grazie sincero a tutta la Cellula Noè, nella quale il cammino è iniziato proprio al momento
dell’ingresso nell’Università e durante tutto questo periodo mi ha accompagnato e sostenuto,
porto sicuro presso cui poter sempre riposare le vele. Insieme a tutti i ragazzi della Parrocchia
di Castel Del Piano, mi avete mostrato cos’è la semplicità del cuore; mentre scoprivo quale
immensità meravigliosa c’è da imparare nei libri e nel creato, un’altra ricchezza mi è stata
donata, ed è quella che viene dalle persone e non si apprende, si vive. Grazie ai “Gemma” per la
musica in cui crediamo e per quello che abbiamo fatto e, soprattutto, faremo. Un grazie speciale
a Don Francesco, perché è stato ed è esempio di Fede e una guida salda e forte, capace di tirar
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fuori il meglio di me.
Questo lavoro e tutti gli anni di studio vissuti fino ad oggi non sarebbero mai stati possibili
senza lo straordinario supporto di tutta la mia famiglia. Parlare di ringraziamenti da parte mia
nei confronti di ciascuno di voi, cari papà e mamma, splendide sorelle Giulia e Costanza e i
tutti i nonni (anche chi non c’è più) è molto più che riduttivo. Siete voi che, più di ogni altro,
giorno dopo giorno mi insegnate il significato di parole come cura, dedizione e amore. E ve ne
sono infinitamente grato. Quello che da sempre fate per me, per permettermi di concentrarmi
sugli studi e mettermi nelle condizioni di riuscirci il meglio possibile, è cosa indescrivibile e non
c’è poi molto altro che io possa dire. L’unico vero modo da parte mia per rispondere a tanti
doni è mettere tutto me stesso in quello che il Signore mi chiama a fare, e spero d’esserci in
parte riuscito, ma più di tutto spero sappiate che ogni lettera scritta (e simbolo!) in questa tesi
e tutto quel poco che ho ottenuto non solo nei miei studi ma in ogni altro impegno in cui mi
sono buttato è davvero, concretamente, in gran parte opera vostra. Grazie.
Ultimo, ma certo non per ordine di importanza, è il grazie a Chi ha voluto tutto questo e l’ha
chiamato a diventare realtà. Perché non ho scelto io dove, come e quando nascere, nè quali
capacità avere. A Colui che conta i miei passi e li guida su sentieri di Luce. A Gesù che è la
Stella il cui Amore brilla nella mia vita più di ogni altra cosa.

The Dirac equation in an external magnetic field in the context

Transcription

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