Fysiikan historia Luento 11

Fysiikan historia
Kevät 2011
Luento 11
Towards quantum
mechanics
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The three main discoveries that paved the way to
quantum mechanics were
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The law of black body radiation (Max Planck 1900)
The quantum theory of electromagnetic radiation (Albert
Einstein 1905)
Atom model (Niels Bohr 1913) [Discussed in the previous
lecture.]
Black body radiation
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Gustav Kirchoff (1824-1887) studied the em spectra
of material and presented the following general rules:
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A hot solid object produces light with a continuous spectrum.
A hot rare gas produces light with spectral lines at discrete
wavelengths (emission spectrum).
A hot solid object surrounded by a cool rare gas (i.e. cooler
than the hot object) produces light with an almost continuous
spectrum which has gaps at discrete wavelengths (absorption
spectrum).
He showed in 1859 that the energy spectrum of the
black body radiation depends only on the
temperature:
c
Eν =
ρ (ν ,T ).
8π
It took decades to find out the functional form of �.
•
Lord Rayleight and James Jeans presented a law that
was good at small frequences but lead to ”infrared
catastrophy” at large frequences:
8πν 2
ρ (ν ,T ) = 3 kT
c
•
Wilhelm Wien’s law was valid at large frequences:
8πν 2 hν
ρ (ν ,T ) = 3 hν / kT
c e
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In 1900 Max Planck invented a function that explained
the spectrum at all frequences:
8πν 2
hν
ρ (ν ,T ) = 3 hν / kT
c e
−1
A couple of months later Planck invented an
interpretation for his law: The electromagnetic field
can absorb and emit em radiation only in integer
multiples of a fundamental unit of energy hν.
h = 6.626× 10−34 Js is Planck’s constant
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Planck did NOT say that the energy of oscillators
(atoms) were discrete, quantized. (Einstein did it in
1909.)
Planck did NOT realize that his theory was
revolutionary and against classical physics. Nobody
thought so – except Einstein.
In 1911-1914 Planck tried to “improve” his theory so
as to make it closer to the classical principles. The
paper was wrong, but the notion of the zero-point
energy was introduced there.
In a sense, it is an unfaithful reconstruction of the
history when one says that Planck invented the
quantization of energy.
The energy spectrum of the cosmic
microwave radiation – the most
exactly measured black body
radiation spectrum.
Quantum theory of light
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In his Annus Mirabilis 1905 Albert Einstein
presented his quantun theory of em radiation: the
electromagnetic field consists of localized particle
like objects with energy hν. These energy quanta do
not decay and they are emitted and absorbed as
such.
As an application this theory he explained the
photoelectric effect and two other effects.
He showed later that Planck’s law can be derived
from his theory.
The photon theory was Einstein’s most important
work. Only after this people started to understand
the importance of Planck’s law, which Einstein
strongly advocated.
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Einstein sovelsi Planckin säteilykaavaa johtaakseen
ominaislämmölle kaavan (1907).
Oli sopusoinnussa Dulongin-Petit’n kaavan kanssa,
kun T on suuri, mutta ominaislämpö pieneni
eksponentiaalisesti, kun T oli pieni.
Holl. Peter Debye tarkensi kaavaa myöhemmin
Vastaminaislämmön selittäminen alkoi käänttää
huomion kvanttifysiikkaan. Walther Nernst innostui
niin, että alkoi organisoida kokousta “teorioiden
uudistamiseksi”. Toteutui 1911 (1. Solvay.-kokous).
Development of Bohr’s theory
•
The atom theory of Niels Bohr was developed in
particular by Arnold Sommerfeld (1868-1951)
Münich. His starting point were action integrals. In
addition to the Bohr’s principal quantum number n
he introduced the orbital quantum number l and the
magnetic quantum number m.
•
This Bohr-Sommerfeld theory explained the Stark
effect (the shifting and splitting of spectral lines of
atoms and molecules due to the presence of an
external static electric field ) and the so called
normal Zeeman effect (the splitting of a spectral line
into several components in the presence of a static
magnetic field ).
•
Soon there appeared new phenomena which the
BS-model could not explain. Bohr extended his
correspondence principle to its extreme to save the
model but eventually the failure was inavoidable.
Sommerfeld was an excellent teacher and supervisor.
Among his doctoral students were four Nobel prize winners
(Werner Heisenberg, Wolfgang Pauli, Peter Debye, and
Hans Bethe), and two of his post graduate students, Linus
Pauling and Isidor I. Rabi, won the prize as well. Many of
these students have educated Nobel prize winners of the
next generation.
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In 1896 Pieter Zeeman (1865-1943) discovered that
spectral lines are split in magnetic field. (Zeeman
effect)
Hendrik Lorentz explained the observation by his
electron theory. The observation showed that
electron are in matter associated to atoms. (The
structure of atoms was still unknown.) Lorentz and
Zeeman obtained the Nobel prize in1902.
The essence of the explanation was precession
caused by the different directions of the angular
velocity and the magnetic field. It was classical
physics.
Zeeman’s photograph of
the split of lines.
Zeeman effect in the Sun.
In 1919 Sommerfeld ja Peter Debey explained the
Zeeman effect with the Bohr-Sommerfeld theory:
angular momentum vector is precessing around the
direction of the magnetic field. As the angular
momentum is quantized, a discrete set of lines appear.
They also predicted the Stark effect (the split of lines
in electric field)
Their theory was, however, unable to explain the
anomalous Zeemen effect – the further splitting of
lines - measured by Albert Michelson and Thomas
Preston in 1898. This effect is due to the spin
precession, but spin was not known at that time.
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Old quantum physics of Bohr and Sommerfeld was
in trouble with many new phenomena: the spectrum
of helium went wrong, the existence of the zero
point energy (Robert Mulliken 1924), Paschen-Back
effect (the anomalous Zeeman effect in large
magnetic fields ) (1912), the result of the SternGerlach experiment etc.
Stern-Gerlach experiment
• Otto Stern wanted to test the quantization of the
angular momentum L in atoms by testing the
quantization of the magnetic moment it would imply.
• He shot atoms through an asymmetric magnetic
field. If the atoms have magnetic moment = 1 Bohr
magneton, the beam should split into three parts as
the magnetic force depends on the direction of the
magnetic moment. Stern and Walther Gerlach so
the splitting of the beam in 1922 using silver atoms.
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Albert Einstein and Paul Ehrenfest showed that the
interaction of magnetic field with atoms is by a factor
1015 too small to explain the result – so the result
was a mystery.
Actually the silver atoms they used have L = 0, and
therefore the magnetic moment is also zero. The
splitting is actually due to the internal angular
momentum, the spin. The spin has only two
quantized values, explaining why they saw just two
lines, not three. (The spin was discovered later in
1926 by Samuel Goudsmit ja George Uhlenbeck. )
Otto Stern (1888-1969)
A post card sent by Gerlach
to Bohr telling of the
discovery.
Walther Gerlach (1889-1979)
The nature of radiation
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Arthur Compton (1892-1967) discovered in 1923
that when electromagnetic waves, eg Röntgen rays,
are scattered by electrons (Compton scattering),
their wavelength is changed exactly as if they were
particles with
p=
hν
, E = hν
c
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Peter Debye (1884-1966) explained the result
theoretically. The result confirmed Einstein’s light
quantum theory.
•
Bohr, who didn’t believe in the light quanta, was
puzzled by the Compton scattering. He had some
desparate explanations: energy is conserved only
statistically, radiation effects are noncausal, ”virtual
oscillators”.
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In 1924 Indian Satyendra Nath Bose (1894-1974)
derived Planck’s radiation law without using
classical physics considering the em radiation as a
gas of photons. Einstein generalized his result to
massive particles. This led to the Bose-Einstein
statistics.
Einstein predicted the existence of what is now
called the Bose-Einstein condensate.
The BE condensate
was experimentally
discovered in 1995
(!) by Eric Cornell
and Carl Wieman.
Quantum mechanics
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In around 1924 it game clear that the old quantum
physics is not the whole story. There were too many
anomalies and unexplained results. Also the logical and
conceptual basis was not satisfactory.
German Max Born (1882-1970): ”The whole conceptual
system of physics should be built on a new basis.”
In1925 German Werner Heisenberg (1901-1976)
published the abstract theory of quantum mechanics.
He followed the positivistic philosophical principle stating
that physics should be expressed through the relations
of (in princible) observable quantities. Therefore the
concepts like the path of electrons in atoms should be
abandoned.
Werner Heisenberg
Max Born
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He replaced the classical Fourier expansions
describing eg the location of an electron in the Bohr’s
state n,
x( n, t ) = ∑ aα ( n) exp[iαω ( n)t ]
α
with
x( n, t ) = ∑ aα ( n, n − α ) exp[iω ( n, n − α )t ]
α
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which describes a transition between the states n
and
n – α, a quantity measured in spectral studies.
The physical observables were represented by the
tables of their values in transitions between the
states. Heisenberg realized that those tables do not
necessary commute, ie AB ≠ BA sometimes. That is,
a measurement of one observable (A) may affect the
values of another observable (B)!
Applied his theory to harmonic oscillator and derived
the zero point energy: En=(n+1/2)hν. Theory could
also explain the anomalous Zeeman effect when spin
was taken into account and the fine structure of the
hydrogen spectrum.
In 1926 Max Born (1882-1970) formulated
Heisenberg’s quantum theory in terms of matrices.
The theory was not accepted by all because it was
not easy to visualize (= it was abstract) and was
based on unfamiliar mathematics.
Matter waves
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A French Louis de Broglie (18921987) presented in his doctoral
thesis in 1924 the hypothesis of the
light-particle dualism: since light
has particle properties (photons),
then particles should have wave
properties.
He postulated:
hν = mc 2 , λ = h / p
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This would mean that Bohr’s orbits
in the hydrogen atom are such that
they allow standing electron matter
waves on them.
A formation of
interference pattern
in a double split
experiment with
electrons. Shows the
wave nature of
particles.
The wave formulation of quantum
mechanics
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A German Edwin Schrödinger (1887-1961)
developed de Broglie’s matter wave idea into
a new formulation of quantum mechanics. In
1926 he published a set of three papers
entitled Quantisierung als Eigenwertproblem.
He started with the classical energy formula
E=p2/2m+V replaced observables with
operators
h ∂
,
p=
2πi ∂x
•
h ∂
E=−
2πi ∂t
Quantization of the values of observables
follow from the requirement that the solutions
of the eigenvalue equation are unique:
HΨ = EΨ
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It was soon shown that Schrödinger’s formulation
(intuitive) and Heisenberg’s formulation (abstract) are
equivalent. Paul Dirac ja Pascual Jordan developed a
general formalism independently of each other later
in 1926.
Heisenberg and Schrödinger didn’t like too much
each other’s formulations:
"I am discouraged, if not repelled by Heisenberg’s theory".
- Erwin Schrodinger (1926) –
"The more I think of the physical part of the Schrödinger
theory, the more detestable I find it. What Schrödinger writes
about visualization makes scarcely any sense, in other words
I think it is sh......
-Werner Heisenberg (8 June 1926) -
It was not very clear to Schrödinger how to interprete
his theory. He talked about the vibrations of electrons in
the atom rather than about matter waves.
Max Born presented the probability interpretation of the
wavefunctions:
ψψ * dV
is the probability to find the particle in the volume
element dV.
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In 1928 Paul Dirac (1902-1984) developed a
relativistic counterpart of the Schrödinger equation,
now called the Dirac equation.
The theory predicted positrons and other
antiparticles. The relativity requirement was not
possible to fulfil otherwise.
Heisenberg and Dirac
The positron was
discovered in 1932 by Carl
Andersson.
Dirac’s theory is an example of the amazing fact that
mathematical theories and full-pure theorists can
reveal facts of Nature.
During the development of quantum mechanics the
focus of physic’s research moved from experimental to
theoretical.
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In 1927 Heisenberg presented an interpretation of
his observation that the operators of different
observables A and B do not necessarily commute,
AB ≠ BA .
The uncertainty princible: The exact values of the
observables A and B cannot be known
simultaneously. Their uncertainties always obey
∆A∆B ≥
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h
4π
Discussions with Niels Bohr were important in
inventing this the most central princible of quantum
mechanics.
The rule is not just a hypothesis but it is built in the
quantum mechanics.
Bohr, Heisenberg and Pauli.
Reactions on QM
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Albert Einstein did not accept the probability
interpretation (”God does not play dice.”). He
invented many gedanken experiments in order to
show flaws in QM. Bohr won the cases one after
another.
In 1935 Einstein, Nathan Podolsky ja Nathan
Rosen presented a famous EPR paradox. It is not a
paradox but a phenomenon (quantum
entanglement) that plays a central role in modern
applications of quantum physics (eg quantum
computing):
Einstein suggested that there are hidden variables that
actually determine the destiny of each particle
separately.
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An Irish John Bell (1928-1990) derived in 1964 so
called Bell’s theorem, showing that one cannot
explain all results of QM with hidden variables. In
other words, if quantum mechanics is correct, the
nature is not locally deterministic.
John Bell with his wife.
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The quantum entanglement is nowadays a well
established phenomenon. An Austrian Anton
Zeilinger (b. 1945) is a leading character in this field.