Download A First Look at Quantum Physics

A First Look at Quantum Physics
A First Look at Quantum Physics
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Historical Note
at the end of the 19th century, the overwhelming success of classical
physics – CM, EM, TD made people believe the ultimate description of
nature has been achieved.
at the turn of the 20th century, classical physics was challenged by
Relativity & microphysics.
the series of breakthroughs：
(1) Max Planck → the energy of a quantum：the energy exchange
between an EM wave & matter occurs only in integer multiples of 
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Historical Note
(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
discrete amounts of
ν.
→
Rutherford’s model
2006 Quantum Mechanics
Bohr’s model
Prof. Y. F. Chen
A First Look at Quantum Physics
Historical Note
(4) Compton → scattering X-rays with e-：the X-ray photons behave
like particles with momenta

c
p
2006 Quantum Mechanics

c
Prof. Y. F. Chen
A First Look at Quantum Physics
Essential Relativity
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
correct forms E  rmc2 , p  rm , T  E  mc 2  (r  1)mc2
where r 
1
1
2
 (1 
2
c
2
)1/ 2
c2
using (1  x)n  1  nx  n(n  1) x 2  n(n  1)(n  2) x3  ...
→ r  (1 

c
2
1/ 2
)
 1
2
2!
1
 ...
2 c2
2
3!
1
2
 p  m & T  m
2
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Essential Relativity
in QM the momentum is a more natural variable than γ, a useful relation
2 2
E 2  (cp)2,the
(mcrest
) energies of various atomic
can be given by
particles will often be quoted in energy units; for the electron and proton
the rest energies are given by
mec2  0.511MeV , mpc2  938.3MeV
the non-relativistic limit of E.g. E 2  (cp)2  (mc 2 )2 , where cp
mc 2
, is
pc 2 1/ 2
p2
p4
2
easily seen to be E  mc (1  ( 2 ) )  mc 

 ...
mc
2m 8m3c 2
2
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Essential Relativity
2
2
2 2
mc 2 , E  (cp)  (mc ) can be
mc 2 2 1/ 2
1 (mc 2 )2
approximated to be E  pc(1  (
) )  pc 
 ... , which is also
pc
2 pc
the ultra-relativistic limit when E
seen to be consistent with the energy-momentum relation for photons,
namely E  pc
(i) e- in atoms：when Z  43 the relativistic effects become non-
negligible.
(ii) deuteron： T  2MeV for the simplest nuclear system; compared with
mp c2  mnc2  939MeV →deuteron can be considered as non-relativistic
system
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
ideal blackbody
spectral distribution only depends
on temperature
spectral energy density of blackbody radiation at different temp.
T1 , T2 , T3 , T1  T2  T3  the peak of the radiation spectrum occurs at freq
that is proportional to the temp.
Wien’s displacement law： max 
2006 Quantum Mechanics
c
max

4.9663
k BT
h
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
blackbody radiation：
(1) Rayleigh’s energy density distribution：
when the cavity is in thermal equilibrium, the EM energy density in 
8 2
to   d is given by u ( , T )  N ( )  E  3  E 
c
according to the equipartition theorem of classical thermodynamics, all
oscillators in the cavity have the same mean energy：

E / K T
E

e
dE

0
 E  
 K BT
E / K T
 e dE
B
B
0
2
8

→ u ( , T )  3 K BT is integrate over all freq, the integral diverges
c
→ this result is absurd → called the ultraviolet catastrophe
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
blackbody radiation：
(2) Plank’s energy density distribution：
avoiding the ultraviolet catastrophe, Planck considered that the energy
exchange between radiation & matter must be discrete：
E  nh

→  E 
 (nh )e
n 0

e
 nh / K BT

 nh / K BT
h
e h / K BT  1
n 0
8 2
→ u ( , T )  3 h / hKT
c e B 1
the spectrum of the blackbody radiation reveals the quantization of
radiation, notably the particle behavior of EM waves
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
photoelectric effect：
K
(1)
Incident light of
energy h
0

Electron ejected with
Kinetic energy
K  h  W
Metal of work function W and threshold freq. 0 
2006 Quantum Mechanics
W
h
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
photoelectric effect：
(2) when a metal is irradiation with light, electrons may get emitted
(3) K  h  W  h(  0 ) it was fond that the magnitude of the
photoelectric current thus generated is proportional to the intensity of
the incident radiation, yet the speed of the electrons does not depend
on the radiation’s intensity, but on its frequency.
→ the photoelectric effect provides compelling evidence for the
corpuscular nature of the EM radiation
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
Compton effects：
Compton treated the incident radiation as a stream of particles-photonscolliding elastically with individual e-
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
Compton effects：
by momentum conservation & energy conservation
→        h (1  cos  )  4c  sin 2 ( )
me c
2
→ the Compton effect confirms that photons behave like particles; they
collide with e- like material particles
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
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：

h
p
,k
p
known as the de Broglie relation, connects the momentum of a particle
with the wavelength & wave vector of the wave corresponding to this
particle
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
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
n  2d sin 
Electron
source
Electron
detector
θ/2
ψ
ψ
d
N: crystal
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantum Physics：
as a Fundamental Constant
wave aspect of particles：
de Broglie’s wavelength：

For an Ni crystal, d  9.1102 nm , 2      sin   cos( )
2
2d
2  9.1102 nm

sin  
cos 25o  16.5 102 nm
n
1
h
hc
(2 ) 197.3MeV 1015 m
 

2
p
2  0.511MeV  54eV
2me c k
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
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
→ Ln
(3-a)
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
(3-b)
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
(3-c)
(3-d)
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
discussion L  n
me v 2 ke2
(i) classically, E 

2
r
2
2
m
v
ke
e
(ii) for a circular orbit, the attractive force = centrifugal force

r
r
2
2 2
mv
m (ke )
(iii) with L  mevr ,  E   e   e
2
2 L2
(iv) considering a transition from L  L1 to L  L2 , according to
me
1
1
(ke 2 )( 2  2 ).
2
L1 L2
& the fractional of angular momentum is so small → h
r
2 2
 (ke )
with L  mevr →
 2 3 3 L
r
me v r
Einstein’s relation, E  h , E  h  
me (ke2 ) 2
L
L3
from (ii),  L 
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
Bohr suggested that L  n  L0 hold even for energy small quantum
number. The allowed value of L 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 L0  0 , then this criterion is satisfied, for L  n
(b) if L0  1
2
1
2
, the allowed values are L  (n  )
(c) with any other value of L0 , however, this condition cannot be met.
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
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
m(ke2 )2
1
1
hf r  2 f r  En  En  En1 
(

)
2
2
2
2
(n  1) n
1
2
(n  1) 2  n 2 (1  ) 2 n 2  3
n
n
m(ke2 )2 1
1
 fr 
( En  En1 ) 

2 2 n3  n
n 1 2
1
a classical particle undergoing circular acceleration would emit radiation
at its orbital freq., which is given by
1
2
En “the connections &
interpolations between the QM & classical description of physical are
stressed in this course.”
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – Bohr Model of H Atom
correspondence principle & the classical period：
(a) show that the correspondence principle can be generalized to show
that the classical periodicity,  , of a quantum system in the large n limit
2
is given by  n 
dEn
dn
(b) using the expression for the quantized energies of a particle in a box
length l , find the classical period  n in state n & compare it to the
expectations based on the classical motion
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – CM & QM
(1) the relationship between CM & QM certain sense is similar to that
which exist between geometric & wave optics
(2) in QM the wave function   A exp( S / h) of quasi-classical form;
where S is called action
(3) the small parameter have is the ratio h / S
transition from QM to CM formally is described by the WKB-method at
h tends to 0
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – CM & QM
the analogy between Optics & Mechanics showed to be vary fruitful to
produce 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
provided to replace the motion of light rays & refractive index with
electron rays and potential, respectively
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Semi-classical model – CM & QM
the 2nd analogy is extended to the wave level, going from Optics to
Mechanics by de Broglie & Schrodinger, obtaining the wave mechanics
& subsequently the QM
from CM to QM：Schrodinger eq. has been recognized as the nonrelativistic limit of a more general wave mechanical formulation induced
by the correspondence with optics. The non-relativistic limit of KleinGordon eq. is just the Schrodinger eq
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantization Rules
Wilson & Sommerfeld offered a scheme that included E  nh , &
as special cases L  n
in essence their scheme consists in quantizing the action variable
J
 pdq
 pdq  nh
of classical mechanics
phase integral
for 1D, p  2m( E  V (q)) , since the particle goes from one limit of
b( E )
oscillation to the other and back J  2 dq 2m( E  V (q))
a(E)
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantization Rules
b
J
b
a
m

2{
2
m
(
E

V
(
q
))}

2{
2
m
(
E

V
(
q
))}

2
dq
→
a 2( E  V (q))
q b E
q  a E
E
the limit of oscillation are determined by E  V .
thus the quantities in the brackets vanish &
b
b
J
dq
1

2

2
dt

T

 period
→


a
a
E
(dq / dt )

2( E  V )
dq
v
m
dt
so if J  nh , then E  nh
the quantization of the action, J, is usually referred to as “the BohrSommerfeld quantum condition.”
2006 Quantum Mechanics
Prof. Y. F. Chen
A First Look at Quantum Physics
Quantization Rules
p2 1
Ex：E 
 m 2 x 2 Harmonic oscillator
2m 2
2
a   2E / m 2 , b  2E / m

if
pdx  2
b
2mE  m 2 2 x 2 dx  4m 
a
b
0
 pdx  nh , then E  nh
b 2  x 2 dx 
2 E


E

, Plank Quantization rule
2
Ex： J   pdq   Ld  nh  L  n
0
for an electron moving in a circular orbit of radius r.
2006 Quantum Mechanics
Prof. Y. F. Chen