A First Look at Quantum Physics
2006 Quantum Mechanics Prof. Y. F. Chen
A First Look at Quantum Physics
at the end of the 19th century, the overwhelming success of classical ph
ysics – CM, EM, TD made people believe the ultimate description of nat
ure has been achieved.
at the turn of the 20th century, classical physics was challenged by Relat
ivity & microphysics.
the series of breakthroughs :
(1) Max Planck → the energy of a quantum : the energy exchange bet
ween an EM wave & matter occurs only in integer multiples of
2006 Quantum Mechanics Prof. Y. F. Chen
Historical Note
A First Look at Quantum Physics
(2) Einstein → photon : light itself is made of discrete bits of energy; an
explanation to the photoelectric problem.
(3) Neils Bohr → model of hydrogen atom : atoms can be found only in
discrete states of energy & atoms with radiation takes place only in discr
ete amounts of ν.
2006 Quantum Mechanics Prof. Y. F. Chen
Historical Note
A First Look at Quantum Physics
→
Rutherford’s model Bohr’s model
(4) Compton → scattering X-rays with e- : the X-ray photons behave lik
e particles with momenta
2006 Quantum Mechanics Prof. Y. F. Chen
Historical Note
A First Look at Quantum Physics
c
pc
for a free particle of rest mass m moving at speed υ, the total energy E,
momentum p, and kinetic energy T can be written in the relativistically c
orrect forms
where
using
→
&
2006 Quantum Mechanics Prof. Y. F. Chen
Essential Relativity
A First Look at Quantum Physics
2 2 2, , ( 1)E rmc p rm T E mc r mc 2
1/ 222
2
1(1 )
1
rc
c
2 3( 1) ( 1)( 2)(1 ) 1 ...
2! 3!n n n n n n
x nx x x
2 2
1/ 22 2
1(1 ) 1 ...
2r
c c
p m 21
2T m
in QM the momentum is a more natural variable than γ, a useful relation
can be given by , the rest energies of various atomic
particles will often be quoted in energy units; for the electron and proton
the rest energies are given by
the non-relativistic limit of E.g. , where , is
easily seen to be
2006 Quantum Mechanics Prof. Y. F. Chen
Essential Relativity
A First Look at Quantum Physics
2 2 2 2( ) ( )E cp mc
2 20.511 , 938.3e pm c MeV m c MeV
2 2 2 2( ) ( )E cp mc 2cp mc2 4
2 2 1/ 2 22 3 2
(1 ( ) ) ...2 8
pc p pE mc mc
mc m m c
the ultra-relativistic limit when , can be approxi
mated to be , which is also seen to
be consistent with the energy-momentum relation for photons, namely
(i) e- in atoms : when the relativistic effects become non-negligib
le.
(ii) deuteron : for the simplest nuclear system; compared with
→deuteron can be considered as non-relativistic sy
stem
2006 Quantum Mechanics Prof. Y. F. Chen
Essential Relativity
A First Look at Quantum Physics
2E mc 2 2 2 2( ) ( )E cp mc 2 2 2
2 1/ 2 1 ( )(1 ( ) ) ...
2
mc mcE pc pc
pc pc
E pc
43Z
2T MeV
2 2 939p nm c m c MeV
spectral energy density of blackbody radiation at different temp.
the peak of the radiation spectrum occurs at freq t
hat is proportional to the temp.
Wien’s displacement law :
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
1 2 3 1 2 3, , , T T T T T T
maxmax
4.9663B
ck T
h
ideal blackbody spectral distribution only depends on temperature
blackbody radiation :
(1) Rayleigh’s energy density distribution :
when the cavity is in thermal equilibrium, the EM energy density in t
o is given by
according to the equipartition theorem of classical thermodynamics, all
oscillators in the cavity have the same mean energy :
→ is integrate over all freq, the integral diverges
→ this result is absurd → called the ultraviolet catastrophe
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
d 2
3
8( , ) ( )u T N E E
c
/
0
/
0
B
B
E K T
BE K T
E e dEE K T
e dE
2
3
8( , ) Bu T K T
c
blackbody radiation :
(2) Plank’s energy density distribution :
avoiding the ultraviolet catastrophe, Planck considered that the energy exc
hange between radiation & matter must be discrete :
→
→
the spectrum of the blackbody radiation reveals the quantization of radiatio
n, notably the particle behavior of EM waves
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
E nh
/
0/
/
0
( )
1
B
BB
nh K T
nh K T
nh K T
n
nh eh
Eee
2
/3
8( , )
1Bh K T
hu T
c e
photoelectric effect :
(1)
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
Incident light of energy h
Electron ejected withKinetic energyK h W
Metal of work function W and threshold freq. 0
W
h
Incident light of energy h
Electron ejected withKinetic energyK h W
Metal of work function W and threshold freq. 0
W
h
K
0
K
0
photoelectric effect : (2) when a
metal is irradiation with light, electrons may get emitted
(3) it was fond that the magnitude of the photoelect
ric current thus generated is proportional to the intensity of the incident ra
diation, yet the speed of the electrons does not depend on the radiation’s i
ntensity, but on its frequency.
→ the photoelectric effect provides compelling evidence for the corpuscul
ar nature of the EM radiation
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
0( )K h W h
Compton effects :
Compton treated the incident radiation as a stream of particles-photons-
colliding elastically with individual e-
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
Compton effects :
by momentum conservation & energy conservation
→
→ the Compton effect confirms that photons behave like particles; they
collide with e- like material particles
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
2(1 cos ) 4 sin ( )2c
e
h
m c
wave aspect of particles :
de Broglie → the wave-particle duality is not restricted to radiation, but
must be universal: all material particles should also display a dual wave-
particle behavior :
known as the de Broglie relation, connects the momentum of a particle
with the wavelength & wave vector of the wave corresponding to this pa
rticle
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
, h p
kp
wave aspect of particles :
Davission-Germer exp. confirmation of de Broglie’s hypothesis :
the intensity max of the scattered e- corresponds to the Bragg formula
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
ψ
Electrondetector
ψ
θ /2
d
Electronsource
N: crystal
ψ
Electrondetector
ψ
θ /2
d
Electronsource
N: crystal
2 sinn d
wave aspect of particles :
de Broglie’s wavelength :
For an Ni crystal, ,
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Physics : as a Fundamental Constant
A First Look at Quantum Physics
29.1 10d nm 2 sin cos( )2
222 2 9.1 10
sin cos 25 16.5 101
od nmnm
n
15
2
(2 ) 197.3 10
2 0.511 542 e
h hc MeV m
p MeV eVm c k
Bohr’s assumption :
(1) only a discrete set of circular stable orbit are allowed
(2) the orbital angular momentum of the electron is an integer multiple of
→
(3-a)
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
L n
(3-b)
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
(3-c)
(3-d)
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
discussion
(i) classically,
(ii) for a circular orbit, the attractive force = centrifugal force
(iii) with ,
(iv) considering a transition from to , according to
Einstein’s relation, , .
& the fractional of angular momentum is so small →
with →
from (ii),
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
L n 2 2
2em v ke
Er
2 2
em v ke
r r
eL m vr2 2 2
2
( )
2 2e em v m ke
EL
1L L 2L L
E h 22 2
1 2
1 1( )( )
2em
E h keL L
2 2
3
( )em kehL
r L
eL m vr2 2
2 3 3
( )
e
keL
r m v r
L
Bohr suggested that hold even for energy small quantum
number. The allowed value of is the same for positive & negative
values, this means that if a given value of the angular momentum is
allowed, its negative must also be allowed.
(a) if , then this criterion is satisfied, for
(b) if , the allowed values are
(c) with any other value of , however, this condition cannot be met.
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
0L n L
L
0 0L L n
0
1
2L
1( )
2L n
0L
correspondence principle first given by Bohr :
Bohr noted that the photons emitted in transitions between the quantized
energy levels satisfy the Balmer formula, written is the form
a classical particle undergoing circular acceleration would emit radiation a
t its orbital freq., which is given by “the connections & interpolati
ons between the QM & classical description of physical are stressed in thi
s course.”
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
2 2
1 2 2 2
( ) 1 12 ( )
2 ( 1)r r n n n
m kehf f E E E
n n
2 2 2 2
3
1 2( 1) (1 )n n n
n n
2 2
1 2 31
1 ( ) 1 1( )
2 2r n nn n
m kef E E
n ��������������
1
2 nE
correspondence principle & the classical period :
(a) show that the correspondence principle can be generalized to show t
hat the classical periodicity, , of a quantum system in the large limit
is given by
(b) using the expression for the quantized energies of a particle in a box
length , find the classical period in state & compare it to the exp
ectations based on the classical motion
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics
n
2n
ndE
dn
l n n
(1) the relationship between CM & QM certain sense is similar to that w
hich exist between geometric & wave optics
(2) in QM the wave function of quasi-classical form; wher
e is called action
(3) the small parameter have is the ratio
transition from QM to CM formally is described by the WKB-method at
tends to
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – CM & QM
A First Look at Quantum Physics
exp( / )A S h
S
/h S
h 0
the analogy between Optics & Mechanics showed to be vary fruitful to p
roduce very important physical insight
the 1st analogy put geometrical optics in correspondence with CM
the development of this analogy was the formulation of electron optics
the formulation of electron optics is similar to EM geometrical optics pro
vided to replace the motion of light rays & refractive index with electron r
ays and potential, respectively
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – CM & QM
A First Look at Quantum Physics
the 2nd analogy is extended to the wave level, going from Optics to Mec
hanics by de Broglie & Schrodinger, obtaining the wave mechanics & su
bsequently the QM
from CM to QM : Schrodinger eq. has been recognized as the non-rela
tivistic limit of a more general wave mechanical formulation induced by t
he correspondence with optics. The non-relativistic limit of Klein-Gordon
eq. is just the Schrodinger eq
2006 Quantum Mechanics Prof. Y. F. Chen
Semi-classical model – CM & QM
A First Look at Quantum Physics
Wilson & Sommerfeld offered a scheme that included , &
as special cases
in essence their scheme consists in quantizing the action variable
of classical mechanics
phase integral
for 1D, , since the particle goes from one limit of oscilla
tion to the other and back
2006 Quantum Mechanics Prof. Y. F. Chen
Quantization Rules
A First Look at Quantum Physics
E nh
L n
J pdqpdq nh
2 ( ( ))p m E V q ( )
( )2 2 ( ( ))
b E
a EJ dq m E V q
→
the limit of oscillation are determined by .
thus the quantities in the brackets vanish &
→
so if , then
the quantization of the action, J, is usually referred to as “the Bohr-Som
merfeld quantum condition.”
2006 Quantum Mechanics Prof. Y. F. Chen
Quantization Rules
A First Look at Quantum Physics
2{ 2 ( ( ))} 2{ 2 ( ( ))} 22( ( ))
b
aq b q a
J b a mm E V q m E V q dq
E E E E V q
E V
2( )E V dqv
m dt
12 2
( / )
b b
a a
J dqdt T period
E dq dt
J nh E nh
Ex : Harmonic oscillator
,
if , then , Plank Quantization rule
Ex :
for an electron moving in a circular orbit of radius r.
2006 Quantum Mechanics Prof. Y. F. Chen
Quantization Rules
A First Look at Quantum Physics
22 21
2 2
pE m x
m
22 /a E m 22 /b E m
2 2 2 2 2
0
22 2 4
b b
a
E Epdx mE m x dx m b x dx
pdx nh E nh
2
0J pdq Ld nh
L n