EP225 Lecture 31 Quantum Mechanical E¤ects 1

EP225 Lecture 31
Quantum Mechanical E¤ects 1
Why the Hydrogen Atom Is Stable
In the classical model of the hydrogen atom, an electron revolves around a proton at
a radius r = 5:3 10 11 m (Bohr radius) and velocity v = 2:2 106 m/sec so that the
centrifugal force mv 2 =r is counterbalanced by the Coulomb force e2 =4 "0 r2 ;
mv 2
e2
=
:
r
4 "0 r 2
The acceleration on the electron is tremendously large,
a=
v2
=9
r
1022 m/s2 ;
and the electron should lose its energy to radiation at a rate
P =
(ea)2
= 2:9
6 "0 c3
1011 eV/s.
If such radiation were allowed, the orbit radius of the electron decreases very quickly (in 50
pico seconds!). The life time of the hydrogen atom should be very short. Of course, this
conclusion is absurd, for the hydrogen atom is known to be very stable. The classical picture
has thus to be modi…ed.
Classical picture of a hydrogen atom.
In quantum mechanical picture, it is postulated that the orbit radius of r = 5:3
m is the possible minimum radius an electron can acquire in the hydrogen atom,
angualr momentum rmv &
h
= ~;
2
where
h = 6:63
= 4:14
10
10
1
34
15
J s
eV s
10
11
is Planck constant. We will return to this important subject later. In quantum mechanics,
the concept of well de…ned particle position and velocity has to be abandoned. Discrete
frequencies are frequently observed in nature. For example, standing waves in a clamped
string are possible only for discrete oscillation frequencies. Such observation strongly suggests
that discrete radiation frequencies from hydrogen atoms should have something to do with
standing waves.
Bohr’s Model of Hydrogen Atom
In the 19th century, spectroscopy as a means to study light emission from various gas
discharges (plasmas) had been well established. The invention of the grating spectrometer
and advancement in vacuum techniques played key roles in spectroscopic studies. Light
emission from hydrogen gas was investigated in particular detail and the data accumulated
prepared a bed for the birth of quantum mechanics. It had been well established that
radiation spectrum from hydrogen gas was discrete and the radiation frequency obeyed the
following empirical formula,
1
1
;
= const.
n2 m2
where n and m are nonzero integers. The observed wavelengths in nm are:
m n = 1 Lyman n = 2 Balmer n = 3 Paschen
2
122
3
103 658 (H-alpha)
4
98 488 (H-beta)
1881
5
95
435
1285
6
94
411
1097
and corresponding energy levels based on Bohr’s model are shown.
Hydrogen atom energy levels.
2
In 1913, Bohr proposed a radical idea that the angular momentum of the electron in the
hydrogen atom is quantized,
rn mvn = n~;
n = 1; 2; 3;
Then, the orbit radius is also quantized as seen from the force balance equation,
mvn2
e2
=
;
rn
4 "0 rn2
(mrn vn )2 = (n~)2 =
me2 rn
;
4 "0
~2 2
n
me2
= 5:3 10 11 (m)
rn = 4 " 0
n2 :
The total energy of a hydrogen atom with electron orbit rn is
1
En = mvn2
2
e2
=
4 " 0 rn
1 e2
;
n 2 8 " 0 r1
where
~2
= 5:3 10 11 m,
me2
is the orbit radius of the ground state (n = 1): The negative energy En < 0 means that an
energy jEn j must be given to the atom to dismantle the atomic bondage due to Coulomb
force. In particular, for n = 1;
r1 = 4 " 0
E1 =
e2
=
8 " 0 r1
13:6 eV,
which means that an energy of 13.6 eV or more must be given to a hydrogen atom at the
ground state to liberate the electron from the proton. This energy is known as the ionization
potential (energy) of hydrogen.
The discrete radiation spectrum from hydrogen discharge can now be understood as
follows. In a gas discharge, ionization of a neutral gas forms a plasma in which electrons and
ions coexist. The discharge itself is maintained by electron bombardment. Electrons acquire
energy from the electric …eld externally applied. In normal discharges with an electron
temperature of order 1 eV (' 11600 K), not all hydrogen atoms are ionized and most atoms
remains neutral but with an energy higher than at the ground state. This process is called
excitation and provides a basic mechanism for laser emission from gases. Excited atoms are
unstable because the electron has room to fall to a lower energy state. If it falls from m-th
to n-th energy state, the energy di¤erence
Emn =
1
n2
1
m2
13:6 eV, m > n;
3
is carried o¤ by radiation (photon) at a frequency
Emn
(Hz).
h
=
Example.
(a) What is the energy of a hydrogen atom excited to n = 3 state? (b) What
is the electron orbit radius of the state? (c) If the atom is deexcited to n = 2 energy level,
what is the wavelength of emitted radiation?
Solution
(a) The energy level of n = 3 state is
E3 =
13:6
= 1:5 eV.
33
(b) The orbit radius is
r3 = n 2 r1
= 9 5:3 10 11 m
= 4:8 10 10 m.
(c) The energy di¤erence is
1 1
4 9
= 1:9 eV.
E =
13:6 eV
The frequency of photon is
E
h
1:9 1:6 10 19 J
=
6:63 10 34 J sec
= 4:6 1014 Hz.
f =
The wavelength is
=
c
= 658 nm.
f
Photoelectric E¤ect
4
(a). Experimental setup of photoelectric e¤ect.
(b). Current vs. voltage in Hertz’s experiment. Vs is the stopping potential,
1 2
eVs = mvmax
=h
2
W:
In 1886, Hertz discovered that certain metals emit electrons when illuminated by light.
The electron was identi…ed much later in 1900 by Thomson and at the time of Hertz’s
experiment, it was not clear what was really carrying electric current which was the quantity
measured in the experiment. It is shown schematically in Fig. 3 (a). When light is on, the
tube becomes a diode with the illuminated metal plate acting as an electron emitting cathode.
The current in the tube disappears when light is o¤ or else short wavelength (ultraviolet)
components are …ltered out. (In the demonstration used in the lecture, recall that if a glass
plate is placed in front of the mercury lamp (strong UV source), nothing happened. Glass
…lters out ultraviolet light e¤ectively.) The amount of current decreases as the negative
voltage increases in magnitude and eventually vanishes at the potential called the stopping
potential Vs as shown in Fig. 3 (b): This may be understood if emitted electrons have a
kinetic energy and at su¢ ciently large potential, all electrons are repelled back to the metal
plate.
A successful theoretical explanation for Hertz’s observation of photoelectric e¤ect was
5
given by Einstein in 1905 in terms of energy conservation,
1
h = mv 2 + W;
2
1
2
where 2 mv is the electron kinetic energy and W is the potential energy which keeps electrons
in the metal under normal conditions. If an electron acquires a su¢ cient energy from the
photon to overcome the potential energy, it is liberated from the metal surface. The work
function W is a constant for a given metal and normally ranges from 2 to 5 eV. Note that
the kinetic energy of electrons
1 2
mv = h
W;
2
is the possible maximum energy an electron an acquire. Some electrons are emitted with
smaller energies between 0 and h
W . The stopping potential is the potential to repel all
electrons. Therefore,
eVs = h
W; or W = h
eVs ;
which allows one to measure the work function W for a given photon energy.
Einstein’s explanation is based on energy conservation. What about momentum conservation? The momentum of photon is
h
h
= ;
pp =
c
in analogy to the momentum of electromagnetic wave (actually any wave),
energy
wave momentum =
:
c
The photon pushes the metal with the momentum pp : Electrons also pushes the metal through
recoil, and the total momentum given to the metal is
h
pp + mv = + mv:
However, the recoil velocity of a metal plate is negligibly small under normal circumstance
and momentum conservation does not a¤ect the basic equation of photoelectric e¤ect. (The
situation is similar to re‡ection analysis of electromagnetic waves at an impedance discontinuity. The formula such as
Z2 Z1
Er =
Ei ;
Z 2 + Z1
has been derived through energy conservation alone.)
Example 2.
Sodium is illuminated by light wave having = 450 m. What are the
maximum electron energy emitted, stopping potential and cuto¤ wavelength ? The work
function of sodium is 2.2 eV.
Solution
From
1 2
mv = h
W
2
c
= h
W
= 4:14
= 2:76
3 108 m/sec
450 10 9 m
2:2 eV = 0:56 eV.
10
15
eV sec
6
2:2 eV
The stopping potential is 0.56 V. The cuto¤ wavelength to cause photoelectric e¤ect is
=
hc
4:14
=
W
10
15
3
2:2
7
108
m = 5:6
10
7
m.