a first look at quantum physics

A First Look at Quantum Physics

2006 Quantum Mechanics Prof. Y. F. Chen


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


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 discr

ete amounts of ν.


Historical Note


→

Rutherford’s model Bohr’s model

(4) Compton → scattering X-rays with e- ： the X-ray photons behave lik

e particles with momenta


Historical Note


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

→

&


Essential Relativity


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




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




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 ：


Quantum Physics ： as a Fundamental Constant


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 　




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




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)




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




0( )K h W h

Compton effects ：

Compton treated the incident radiation as a stream of particles-photons-

colliding elastically with individual e- 　




Compton effects ：

by momentum conservation & energy conservation

→

→ the Compton effect confirms that photons behave like particles; they

collide with e- like material particles




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




, h p

kp


Davission-Germer exp. confirmation of de Broglie’s hypothesis ：

the intensity max of the scattered e- corresponds to the Bragg formula




ψ

Electrondetector

ψ

θ /2

d

Electronsource

N: crystal

ψ

Electrondetector

ψ

θ /2

d

Electronsource

N: crystal

2 sinn d


de Broglie’s wavelength ：

For an Ni crystal, ,




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)

　


Semi-classical model – Bohr Model of H Atom


L n

(3-b)

　




(3-c)

(3-d)

　




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),




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.




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.”




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




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


Semi-classical model – CM & QM


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




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




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


Quantization Rules


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.”


Quantization Rules


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.


Quantization Rules


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

a first look at quantum physics

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