Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator.

Quantum Harmonic Oscillator

2006 Quantum Mechanics Prof. Y. F. Chen


1D S.H.O. ： linear restoring force , k is the force constant

& parabolic potential

.

harmonic potential’s minimum at = a point of stability in a system




xkxF )(

2/)( 2xkxV

A particle oscillating in a harmonic potential

0x

Ex ： the positions of atoms that form a crystal are stabilized by the pre

sence of a potential that has a local min at the location of each atom

→

∵ the atom position is stabilized by the potential, a local min results in th

e first derivative of the series expansion = 0

∴

→ a local min in V(x) is only approximated by the quadratic function of a

H.O.




0

)()(

!

1)(

n

no

xxn

n

xxdx

xVd

nxV

o

2

2

2

)()(

2

1)(

)()()( o

xx

oxx

o xxdx

xVdxx

dx

xdVxVxV

oo

2

2

2

)()(

2

1)()( o

xx

o xxdx

xVdxVxV

o

for the H.O. potential 　　　　　　　　 , the time-indep Schrödinger wa

ve eq. ：

use(1) & (2)

→

making the substitution

→ called Hermite functions.


Schrödinger Wave Eq. for 1D Harmonic Oscillator


2/)( 22 xmxV

)()(2

1

222

2

22

xExxmxd

d

mnnn

xm

n

n

E2

0)(~)(~2

2

2

nn

n

d

d

)()(~ 2/2

nn He

0)(1)(

2)(

2

2

nnnn Hd

dH

d

Hd

One important class of orthogonal polynomials encountered in QM & las

er physics is the Hermite polynomials, which can be defined by the form

ula

the first few Hermite polynomials are ：

in general ：　　

.


Hermite Functions


,2,1,0,)1()(2

2

nd

edeH

n

nn

n

128)(,24)(,2)(,1)( 33

2210 HHHH

knn

n

k

n knk

nH 2

]2/[

0

)2()!2(!

!)1()(

the Hermite polynomials come from the generating function ：

　　　　　　　　　　　　　　　　　 .

→ Taylor series ：

　　　　　　　　　　　　　　　　　　 .

→

substituting into 　：

→ recurrence relation ：


Hermite Functions


tn

tHetg

n

nn

tt ,!

)(),(0

22

tt

g

n

tetg

n tn

nntt ,

!),(

0 0

22

)()1(2

222

0

)(

0

n

u

n

unn

t

tn

n

tn

n

Hud

edee

te

t

g

2 2

0

( , ) ( )!

nt t

nn

tg t e H

n

gtt

g)22(

,2,1,)(2)(2)( 11 nHnHH nnn

substituting into 　：

→ recurrence relation ：

with &

→ 2nd-order ordinary differential equation for

eigenvalues of the 1D quantum H.O. ：


Hermite Functions


2 2

0

( , ) ( )!

nt t

nn

tg t e H

n

gtx

g2

1

00 !

)(2

!

)(

n

n

nn

n

n tn

Ht

n

H

,2,1,)(2)(

1 nHnd

dHn

n

1 1( ) 2 ( ) 2 ( )n n nH H n H 1

( )2 ( )n

n

dHn H

d

)(nH

0)(2)(

2)(

2

2

n

nn Hnd

dH

d

Hd

2

112 nEn nn

the eigenfunctions of 1D H.O. ：

with the help of , find normalization consta

nt , →

(i) in CM, the oscillator is forbidden to go beyond the potential, beyond t

he turning points where its kinetic energy turns negative.

(ii) the quantum wave functions extend beyond the potential, and thus th

ere is a finite probability for the oscillator to be found in a classically forb

idden region 　


Stationary States of 1D Harmonic Oscillator


)()(~ 2/2

nnn HeC

!2)( 22

ndHe nn

nC

!2)( 22

ndHe nn

n=0 n=1

n=2 n=3

n=4 n=5

n

n=0 n=1

n=2 n=3

n=4 n=5

n=0 n=1

n=2 n=3

n=4 n=5

n=0 n=1

n=2 n=3

n=4 n=5

n




the classical probability of finding the particle inside a region 　　：

.

the velocity can be expressed as a function of 　：

→




2 / ( )( )

2 /cl

t vP

T

( ) sin ( )v A t

22)( Av

2 2

1 1( )clP

A

(i) the difference between the two probabilities for n=0 is extremely

striking there is no zero-point energy in CM ∵

(ii) the quantum and classical probability distributions coincide when the

quantum number n becomes large

(iii) this is an evidence of Bohr’s correspondence principle




n=0 n=30n=30

(1) classically, the motion of the H.O. is in such a manner that the positi

on of the particle changes from one moment to another.

(2) however, although there is a probability distribution for any eigenstat

e in QM, this distribution is indep of time → stationary states

(3) even so, the Ehrenfest theorem reveals that a coherent superpositio

n of a number of eigenstates, i.e., so-called “wave packet state”, will lea

d to the classical behavior




show ：

using the generation function , we can have

∵ the orthogonality property, the integration leads to

→

as a consequence, we can obtain




!2)( 22

ndHe nn

0 0

22

!!)()(

2222

m

mn

nmn

sstt

mn

stHHeeee

0

222)( )(!!

22

nn

nnststts dHe

nn

stedee

0

2

0

)(!!!

2 2

nn

nn

n

n

dHenn

st

n

ts

!2)( 22

ndHe nn

given a mean rate of occurrence r of the events in the relevant interval,

the Poisson distribution gives the probability that exactly n

events will occur

for a small time interval the probability of receiving a call is .

the probability of receiving no call during the same tiny interval is

given by . the probability of receiving exactly n calls in the total

interval is given by


The Poisson Distribution


)( nXP

t tr

t

tr1

tt

trtPtrtPttP nnn )(1)()( 1

rearranging , dividing through by ,

and letting , the differential recurrence eq. can be found and writt

en as

for ：

which can be integrated to lead to

with the fact that the probability of receiving no calls in a zero time

interval must be equal to unity ：




trtPtrtPttP nnn )(1)()( 1 t

0 t

)()()(

1 tPrtPrtd

tdPnn

n

0n )()(

00 tPrtd

tdP

trePtP )0()( 00

)0(0P

tretP )(0

substituting into for ：

, repeating this process, can be found to be

the sum of the probabilities is unity ：

the mean of the Poisson distribution ：　




tretP )(0 )()()(

1 tPrtPrtd

tdPnn

n 1n

tretrtP )()(1 )(tPn

( )( )

!

nr t

n

r tP t e

n

1!

)(

!

)()(

000

trtr

n

ntr

n

trn

nn ee

n

tree

n

trtP

rtn

rttree

n

trntnPn

n

ntr

n

trn

nn

1

1

00 !)1(

)()(

!

)()(

in other words, the Poisson distribution with a mean of is given by ：




en

Pn

n!

)(

The Schrödinger coherent wave packet state can be generalized as

with

it can be found that the norm square of the coefficient is exactly

the same as the Poisson distribution with the mean of


Schrödinger Coherent States of the 1D H.O.


0

)(~),(n

tEi

nn

n

ect

2/2

!

)( en

ec

ni

n

2|| nc

2

substituting & into

using




1

2nE n

)(!2)(~ 2/2/1 2

n

nn Hen

0

( , ) ( ) :nEi t

n nn

t c e

2 2

2 2

/ 2 / 2 ( 1/ 2)

0

( )

( ) / 2 / 21/ 4

0

( ) 1( , ) ( )

! 2 !

/ 21 ( )

!

i ni n t

nnn

ni t

i tn

n

et e H e e

n n

ee e H

n

2 2

0

( , ) ( ) :!

nt t

nn

tg t e H

n

2 2

2 2

2( ) / 2 / 2 ( ) ( )

1/ 4

( ) / 2 / 2 2 2( ) ( )1/ 4

1( , ) exp / 2 2

1 exp / 2 2

i t i t i t

i t i t i t

t e e e e

e e e e

as a result, the probability distribution of the coherent state is given by ：

it can be clearly seen that the center of the wave packet moves in the p

ath of the classical motion




2 2( ) 2

2 2 2

2

1( , ) ( , ) ( , ) exp cos[2( )] 2 2 cos( )

1 exp 2 cos ( ) 2 2 cos( )

1 exp [ 2 cos( )]

P t t t e t t

t t

t

)cos(2 t

with , &

the operator acting on the eigenstate


Creation & Annihilation Operators


xm

1 1( ) 2 ( ) 2 ( )n n nH H nH 21/ 2

/ 2( ) 2 ! ( )nn nn e H

x )(~ n

)(~)(~12

1

)()(2

1!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/1

2/2/1

2

2

2

nn

nnn

nn

nn

n

nnm

HnHenm

Henm

Henm

x

in a similar way, the operator acting on the eigenstate




)(~ nxp

)(~)(~12

1

)()(2

1!2

)()()(!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/2/1

2/2/1

2/2/1

2

22

2

2

nn

nnn

nnn

nn

nn

nx

nnmi

HnHenmi

HeHenmi

Henmi

Henx

ip

→

&

consequently, it is convenient to define 2 new operators ：

&




)(~1)(~ˆ1

ˆ2

11

nnx np

mix

m

)(~)(~ˆ1

ˆ2

11

nnx np

mix

m

xp

mix

ma ˆ

1ˆ

2

1ˆ†

xp

mix

ma ˆ

1ˆ

2

1ˆ

the operator is the increasing (creation) operator ：

this means that operating with on the n-th stationary states yields a s

tate, which is proportional to the higher (n +1)-th state

the operator is the lowering (annihilation) operator ：


tate, which is proportional to the higher (n -1)-th state 　




†a

)(~1)(~ˆ 1† nn na

†a

a

)(~)(~ˆ 1 nn na

a

in terms of & , the operators & can be expressed as ：

&

we can find the commutator of these 2 ladder operators ：

which is the so-called canonical commutation relation




a †a x xp

†ˆˆ2

ˆ aam

x

†ˆˆ2

ˆ aam

ipx

1ˆ,ˆˆ,ˆ2

1

ˆ1

ˆ,ˆ1

ˆ2

1]ˆ,ˆ[ †

xpi

pxi

pm

ixm

pm

ixm

aa

xx

xx

is the hermitian conjugate 　：

proof ：




†a a

1

†221 |ˆ||ˆ| aa

1 2 1 2

1 2 1 2

2 1 2 1

2 1

1 1ˆ ˆ ˆ| |

2

1 1ˆ ˆ

2

1 1ˆ ˆ

2

1 1ˆ ˆ

2

x

x

x

x

ma x i p

m

mx i p

m

mx i p

m

mx i p

m

†2 1ˆ | |a

with , &

the operator acting on the eigenstate



xm

1 1( ) 2 ( ) 2 ( )n n nH H nH 21/ 2

/ 2( ) 2 ! ( )nn nn e H

x )(~ n

)(~)(~12

1

)()(2

1!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/1

2/2/1

2

2

2

nn

nnn

nn

nn

n

nnm

HnHenm

Henm

Henm

x


in a similar way, the operator acting on the eigenstate



)(~ nxp

)(~)(~12

1

)()(2

1!2

)()()(!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/2/1

2/2/1

2/2/1

2

22

2

2

nn

nnn

nnn

nn

nn

nx

nnmi

HnHenmi

HeHenmi

Henmi

Henx

ip


→

&

consequently, it is convenient to define 2 new operators ：

&



)(~1)(~ˆ1

ˆ2

11

nnx np

mix

m

)(~)(~ˆ1

ˆ2

11

nnx np

mix

m

xp

mix

ma ˆ

1ˆ

2

1ˆ†

xp

mix

ma ˆ

1ˆ

2

1ˆ


the operator is the increasing (creation) operator ：


tate, which is proportional to the higher (n +1)-th state

the operator is the lowering (annihilation) operator ：


tate, which is proportional to the higher (n -1)-th state 　



†a

)(~1)(~ˆ 1† nn na

†a

a

)(~)(~ˆ 1 nn na

a


in terms of & , the operators & can be expressed as ：

&

we can find the commutator of these 2 ladder operators ：

which is the so-called canonical commutation relation



a †a x xp

†ˆˆ2

ˆ aam

x

†ˆˆ2

ˆ aam

ipx

1ˆ,ˆˆ,ˆ2

1

ˆ1

ˆ,ˆ1

ˆ2

1]ˆ,ˆ[ †

xpi

pxi

pm

ixm

pm

ixm

aa

xx

xx


is the hermitian conjugate 　：

proof ：



†a a

1

†221 |ˆ||ˆ| aa

1 2 1 2

1 2 1 2

2 1 2 1

2 1

1 1ˆ ˆ ˆ| |

2

1 1ˆ ˆ

2

1 1ˆ ˆ

2

1 1ˆ ˆ

2

x

x

x

x

ma x i p

m

mx i p

m

mx i p

m

mx i p

m

†2 1ˆ | |a


with

&

→

using the commutation relation

→

define the so-called number operator ：

→ the H.O. Hamiltonian takes the form ：



††††††2

ˆˆˆˆˆˆˆˆ4

ˆˆˆˆ42

ˆaaaaaaaaaaaa

m

px

††††††22 ˆˆˆˆˆˆˆˆ4

ˆˆˆˆ4

ˆ2

1aaaaaaaaaaaaxm

aaaaxmm

pH x ˆˆˆˆ

2ˆ

2

1

2

ˆˆ ††222

1ˆˆˆˆ]ˆ,ˆ[ ††† aaaaaa

2

1ˆˆˆ †aaH

aaN ˆˆˆ †

2

1ˆˆ NH


the eigenstates of can be found to be coherent states ：

coherent states have the minimum uncertainty




a );0,(

0

2/||0

ˆ2/|| )(~!

)(~);0,(2†2

nn

na

neee

†ˆ ˆ ˆ( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

2 ( ) cos

2

i x a am

m m

22 †

2 2

ˆ ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( 1)2

x a am

m

22 2ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

x x xm

as a consequence, we obtain the minimum uncertainty state ：




†ˆ ˆ ˆ( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( ) 2 sin2

x

mii p i a a

mi m

22 †

2 2

ˆ ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( 1)2

x

mp a a

m

2);0,(ˆ);0,();0,(ˆ);0,( 222 m

ppp xxx

2

xpx

Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator.

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Transcript of Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator.