Quantum Harmonic Oscillator - National Chiao Tung Universityocw.nctu.edu.tw/course/physics/quantummechanics_lecturenotes/chapter05.pdf · Quantum Harmonic Oscillator. 2006 Quantum

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


Quantum Harmonic OscillatorQuantum Harmonic Oscillator

xkxF −=)(

2/)( 2xkxV =

A particle oscillating in a harmonic potential

0=x

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

presence 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

the first derivative of the series expansion = 0

∴

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

H.O.


Quantum Harmonic OscillatorQuantum Harmonic Oscillator

∑∞

= −

−=0

)()(!

1)(n

no

xxn

n

xxdx

xVdn

xVo

L+−+−+=−−

22

2

)()(21)()()()( o

xxo

xxo xx

dxxVdxx

dxxdVxVxV

oo

L+−+=−

22

2

)()(21)()( o

xxo xx

dxxVdxVxV

o

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

wave eq.：

use(1) & (2)

→

making the substitution

→

called Hermite functions.


Schrödinger Wave Eq. for 1D Harmonic OscillatorQuantum Harmonic Oscillator

2/)( 22 xmxV ω=

)()(21

222

2

22

xExxmxd

dm nnn ψψω =⎥

⎦

⎤⎢⎣

⎡+−

h

xmh

ωξ =ω

εh

nn

E2=

( ) 0)(~)(~2

2

2

=−+ ξψξεξξψ

nnn

dd

)()(~ 2/2

ξξψ ξnn He−=

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

2

=−+− ξεξξξ

ξξ

nnnn H

ddH

dHd

One important class of orthogonal polynomials encountered in QM &

laser physics is the Hermite polynomials, which can be defined by the

formula

the first few Hermite polynomials are：

in general：

.


Hermite FunctionsQuantum Harmonic Oscillator

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

2

=−=−

ndedeH n

nn

n ξξ

ξξ

ξξξξξξξξ 128)(,24)(,2)(,1)( 33

2210 −=−=== HHHH

knn

n

k

n knknH 2

]2/[

0

)2()!2(!

!)1()( −

=∑ −

−= ξξ

the Hermite polynomials come from the generating function：

.

→ Taylor series：

.

→

substituting into ：

→ recurrence relation：



∞<== ∑∞

=

+− tntHetg

n

nn

tt ,!

)(),(0

22

ξξ ξ

∞<∂∂

== ∑∞

= =

+− ttg

ntetg

n tn

nntt ,

!),(

0 0

22 ξξ

)()1(2

222

0

)(

0

ξξ

ξξξn

un

unn

t

tn

n

tn

n

Hudedee

te

tg

≡−=∂∂

=∂∂

=

−

=

−−

=

2 2

0

( , ) ( )!

nt t

nn

tg t e Hn

ξξ ξ∞

− +

=

= =∑ gttg )22( −=

∂∂ ξ

L,2,1,)(2)(2)( 11 =−= −+ nHnHH nnn ξξξξ

substituting into ：

→ recurrence relation：

with &

→ 2nd-order ordinary differential equation for

eigenvalues of the 1D quantum H.O.：



2 2

0

( , ) ( )!

nt t

nn

tg t e Hn

ξξ ξ∞

− +

=

= =∑ gtxg 2=

∂∂

1

00 !)(2

!)( +

∞

=

∞

=∑∑ =

′ n

n

nn

n

n tn

Htn

H ξξ

L,2,1,)(2)(1 == − nHn

ddH

nn ξξξ

1 1( ) 2 ( ) 2 ( )n n nH H n Hξ ξ ξ ξ+ −= − 1( ) 2 ( )n

ndH n H

dξ ξξ −=

)(ξnH

0)(2)(2)(2

2

=+− ξξξ

ξξξ

nnn Hn

ddH

dHd

ωε h⎟⎠⎞⎜

⎝⎛ +=⇒+=

2112 nEn nn

the eigenfunctions of 1D H.O.：

with the help of , find normalization

constant , →

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

the turning points where its kinetic energy turns negative.

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

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

forbidden region


Stationary States of 1D Harmonic OscillatorQuantum Harmonic Oscillator

)()(~ 2/2

ξξψ ξnnn HeC −=

[ ] πξξξ ⋅=∫∞

∞−

− !2)( 22

ndHe nn

nC

[ ] πξξξ ⋅=∫∞

∞−

− !2)( 22

ndHe nn

n=0 n=1

n=2 n=3

n=4 n=5

ξξ

( )ξψ n

n=0 n=1

n=2 n=3

n=4 n=5

ξξ

n=0 n=1

n=2 n=3

n=4 n=5

n=0 n=1

n=2 n=3

n=4 n=5

ξξ

( )ξψ n



the classical probability of finding the particle inside a region ：

.

the velocity can be expressed as a function of ：

→



ξΔ

2 / ( )( )2 /cl

t vPT

ξ ξξ ξπ ω

Δ ΔΔ = =

( ) sin ( )v A tξ ω ω= ξ

( )22)( ξωξ −= Av

( )2 2

1 1( )clPA

ξ ξ ξπ ξ

Δ = Δ−

(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

position of the particle changes from one moment to another.

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

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

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

superposition of a number of eigenstates, i.e., so-called “wave packet

state”, will lead 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



[ ] πξξξ ⋅=∫∞

∞−

− !2)( 22

ndHe nn

∑∑∞

=

∞

=

−+−+−− =⋅⋅0 0

22

!!)()(

2222

m

mn

nmn

sstt

mnstHHeeee ξξξξξξ

[ ]∑ ∫∫∞

=

∞

∞−

−∞

∞−

−−− =⋅=⋅0

222)( )(!!

22

nn

nnststts dHe

nnstedee ξξπξ ξξ

( ) [ ]∑ ∫∑∞

=

∞

∞−

−∞

=

=⋅0

2

0

)(!!!

2 2

nn

nn

n

n

dHennst

nts ξξπ ξ

[ ] πξξξ ⋅=∫∞

∞−

− !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 DistributionQuantum Harmonic Oscillator

)( nXP =

tΔ trΔ

tΔ

trΔ−1

tt Δ+

( ) trtPtrtPttP nnn Δ+Δ−=Δ+ − )(1)()( 1

rearranging , dividing through by ,

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

written 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 Δ+Δ−=Δ+ − )(1)()( 1 tΔ

0→Δ t

)()()(

1 tPrtPrtdtdP

nnn −= −

0=n )()(

00 tPrtdtdP

−=

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 tPrtPr

tdtdP

nnn −= − 1=n

tretrtP −= )()(1 )(tPn

( )( )!

nr t

nr tP t en

−=

1!)(

!)()(

000=⋅=== −

∞

=

−∞

=

−∞

=∑∑∑ trtr

n

ntr

n

trn

nn ee

ntree

ntrtP

rtn

rttreentrntnPn

n

ntr

n

trn

nn =

−==>=< ∑∑∑

∞

=

−−

∞

=

−∞

= 1

1

00 !)1()()(

!)()(

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



λλλ −= 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.Quantum Harmonic Oscillator

∑∞

=

−=Ψ

0)(~),(

n

tEi

nn

n

ect hξψξ

2/2

!)( α

φα −= en

ecni

n

2|| nc

2α

substituting & into

using



12nE n ω⎛ ⎞= +⎜ ⎟

⎝ ⎠h ( ) )(!2)(~ 2/2/1 2

ξπξψ ξn

nn Hen −−

⋅=

0( , ) ( ) :

nEi t

n nn

t c eξ ψ ξ∞ −

=

Ψ =∑ h%

2 2

2 2

/ 2 / 2 ( 1/ 2)

0

( )( ) / 2 / 2

1/ 40

( ) 1( , ) ( )! 2 !

/ 21 ( )!

i ni n t

nnn

ni ti t

nn

et e H e en n

ee e H

n

φα ξ ω

ω φα ξ ω

αξ ξπ

αξ

π

∞− − − +

=

− −∞

− + −

=

Ψ =

⎡ ⎤⎣ ⎦=

∑

∑

2 2

0( , ) ( ) :

!

nt t

nn

tg t e Hn

ξξ ξ∞

− +

=

= =∑

{ }{ }

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

α ξ ω ω φ ω φ

α ξ ω ω φ ω φ

ξ α α ξπ

α α ξπ

− + − − − − −

− + − − − − −

⎡ ⎤Ψ = − +⎣ ⎦

= − +

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

path 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

α ξξ ξ ξ α ω φ αξ ω φπ

ξ α ω φ α ξ ω φπ

ξ α ω φπ

∗ − += Ψ Ψ = − − + −

= − − − + −

= − − −

)cos(2 φωαξ −= t

with , &

the operator acting on the eigenstate


Creation & Annihilation OperatorsQuantum Harmonic Oscillator

xmh

ωξ = 1 1( ) 2 ( ) 2 ( )n n nH H nHξ ξ ξ ξ+ −= − ( ) 21/ 2/ 2( ) 2 ! ( )n

n nn e Hξψ ξ π ξ−

−=%

x )(~ ξψ n

( )

( )

( )

[ ])(~)(~12

1

)()(21!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/1

2/2/1

2

2

2

ξψξψω

ξξπω

ξξπω

ξπξω

ξψ

ξ

ξ

ξ

−+

−+−−

−−

−−

++⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎦⎤

⎢⎣⎡ +⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

nn

nnn

nn

nn

n

nnm

HnHenm

Henm

Henm

x

h

h

h

h

in a similar way, the operator acting on the eigenstate



)(~ ξψ nxp

( )

( )

( ) ( ) [ ]

( ) ( )

( ) [ ])(~)(~12

1

)()(21!2

)()()(!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/2/1

2/2/1

2/2/1

2

22

2

2

ξψξψω

ξξπω

ξξξπω

ξπξ

ω

ξπξψ

ξ

ξξ

ξ

ξ

−+

−+−−

−−−

−−

−−

−+=

⎥⎦⎤

⎢⎣⎡ −⋅=

′+−⋅−=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−=

nn

nnn

nnn

nn

nn

nx

nnmi

HnHenmi

HeHenmi

Henmi

Henx

ip

h

h

h

h

h

→

&

consequently, it is convenient to define 2 new operators：

&



)(~1)(~ˆ1ˆ2

11 ξψξψ

ωω

++=⎟⎟⎠

⎞⎜⎜⎝

⎛− nnx np

mixm

hh

)(~)(~ˆ1ˆ2

11 ξψξψ

ωω

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+ nnx np

mixm

hh

⎟⎟⎠

⎞⎜⎜⎝

⎛−= xp

mixma ˆ1ˆ

21ˆ†

hh ωω

⎟⎟⎠

⎞⎜⎜⎝

⎛+= xp

mixma ˆ1ˆ

21ˆ

hh ωω

the operator is the increasing (creation) operator：

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

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

the operator is the lowering (annihilation) operator：


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



†a

)(~1)(~ˆ 1† ξψξψ ++= nn na

†a

a

)(~)(~ˆ 1 ξψξψ −= 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 +=ωh ( )†ˆˆ

2ˆ aamipx −−=

ωh

[ ] [ ] 1ˆ,ˆˆ,ˆ21

ˆ1ˆ,ˆ1ˆ21]ˆ,ˆ[ †

=⎟⎠⎞

⎜⎝⎛ +−

=

⎥⎦

⎤⎢⎣

⎡−+=

xpipxi

pm

ixmpm

ixmaa

xx

xx

hh

hhhh ωω

ωω

is the hermitian conjugate ：

proof：



†a a∗

= 1†

221 |ˆ||ˆ| ψψψψ aa

1 2 1 2

1 2 1 2

2 1 2 1

2 1

1 1? �| |2

1 1? 2

1 1? 2

1 1? 2

x

x

x

x

ma x i pm

m x i pm

m x i pm

m x i pm

ωψ ψ ψ ψω

ω ψ ψ ψ ψω

ω ψ ψ ψ ψω

ωψ ψω

∗ ∗

∗

= +

⎡ ⎤= +⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= +⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

h h

h h

h h

h h

†2 1ˆ | |aψ ψ

∗=

with , &

the operator acting on the eigenstate


Creation & Annihilation Operators

xmh

ωξ = 1 1( ) 2 ( ) 2 ( )n n nH H nHξ ξ ξ ξ+ −= − ( ) 21/ 2/ 2( ) 2 ! ( )n

n nn e Hξψ ξ π ξ−

−=%

x )(~ ξψ n

( )

( )

( )

[ ])(~)(~12

1

)()(21!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/1

2/2/1

2

2

2

ξψξψω

ξξπω

ξξπω

ξπξω

ξψ

ξ

ξ

ξ

−+

−+−−

−−

−−

++⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎦⎤

⎢⎣⎡ +⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

nn

nnn

nn

nn

n

nnm

HnHenm

Henm

Henm

x

h

h

h

h


in a similar way, the operator acting on the eigenstate



)(~ ξψ nxp

( )

( )

( ) ( ) [ ]

( ) ( )

( ) [ ])(~)(~12

1

)()(21!2

)()()(!2

)(!2

)(!2)(~ˆ

11

112/2/1

2/2/2/1

2/2/1

2/2/1

2

22

2

2

ξψξψω

ξξπω

ξξξπω

ξπξ

ω

ξπξψ

ξ

ξξ

ξ

ξ

−+

−+−−

−−−

−−

−−

−+=

⎥⎦⎤

⎢⎣⎡ −⋅=

′+−⋅−=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−=

nn

nnn

nnn

nn

nn

nx

nnmi

HnHenmi

HeHenmi

Henmi

Henx

ip

h

h

h

h

h


→

&

consequently, it is convenient to define 2 new operators：

&



)(~1)(~ˆ1ˆ2

11 ξψξψ

ωω

++=⎟⎟⎠

⎞⎜⎜⎝

⎛− nnx np

mixm

hh

)(~)(~ˆ1ˆ2

11 ξψξψ

ωω

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+ nnx np

mixm

hh

⎟⎟⎠

⎞⎜⎜⎝

⎛−= xp

mixma ˆ1ˆ

21ˆ†

hh ωω

⎟⎟⎠

⎞⎜⎜⎝

⎛+= xp

mixma ˆ1ˆ

21ˆ

hh ωω


the operator is the increasing (creation) operator：


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

the operator is the lowering (annihilation) operator：


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


Creation & Annihilation Operators†a

)(~1)(~ˆ 1† ξψξψ ++= nn na

†a

a

)(~)(~ˆ 1 ξψξψ −= 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 +=ωh ( )†ˆˆ

2ˆ aamipx −−=

ωh

[ ] [ ] 1ˆ,ˆˆ,ˆ21

ˆ1ˆ,ˆ1ˆ21]ˆ,ˆ[ †

=⎟⎠⎞

⎜⎝⎛ +−

=

⎥⎦

⎤⎢⎣

⎡−+=

xpipxi

pm

ixmpm

ixmaa

xx

xx

hh

hhhh ωω

ωω


is the hermitian conjugate ：

proof：


Creation & Annihilation Operators†a a

∗= 1

†221 |ˆ||ˆ| ψψψψ aa

1 2 1 2

1 2 1 2

2 1 2 1

2 1

1 1? �| |2

1 1? 2

1 1? 2

1 1? 2

x

x

x

x

ma x i pm

m x i pm

m x i pm

m x i pm

ωψ ψ ψ ψω

ω ψ ψ ψ ψω

ω ψ ψ ψ ψω

ωψ ψω

∗ ∗

∗

= +

⎡ ⎤= +⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= +⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

h h

h h

h h

h h

†2 1ˆ | |aψ ψ

∗=


with

&

→

using the commutation relation

→

define the so-called number operator：

→ the H.O. Hamiltonian takes the form：



( ) ( ) ( )††††††2

ˆˆˆˆˆˆˆˆ4

ˆˆˆˆ42

ˆaaaaaaaaaaaa

mpx −−+=−−−=

ωω hh

( )( ) ( )††††††22 ˆˆˆˆˆˆˆˆ4

ˆˆˆˆ4

ˆ21 aaaaaaaaaaaaxm +++=++=

ωωω hh

( )aaaaxmm

pH x ˆˆˆˆ

2ˆ

21

2ˆˆ ††22

2

+=+=ωω h

1ˆˆˆˆ]ˆ,ˆ[ ††† =−= aaaaaa

⎟⎠⎞

⎜⎝⎛ +=

21ˆˆˆ †aaH ωh

aaN ˆˆˆ †=

⎟⎠⎞

⎜⎝⎛ +=

21ˆˆ NH ωh


the eigenstates of can be found to be coherent states ：

coherent states have the minimum uncertainty



a );0,( αξΨ

∑∞

=

−− ==Ψ0

2/||0

ˆ2/|| )(~!

)(~);0,(2†2

nn

na

neee ξψαξψαξ ααα

†? �( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

2 ( ) cos2

i x a am

m m

ξ α ξ α ξ α ξ αω

α α φω ω

∗

Ψ Ψ = Ψ + Ψ

= + =

h

h h

( )22 �

2 2

? �( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( 1)2

x a am

m


α α α α α αω

∗ ∗ ∗

Ψ Ψ = Ψ + Ψ

⎡ ⎤= + + + +⎣ ⎦

h

h

22 2?( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

x x xm


→ Δ = Ψ Ψ − Ψ Ψ =h

as a consequence, we obtain the minimum uncertainty state：



†?�( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( ) 2 sin2

xmii p i a a

mi m

ωξ α ξ α ξ α ξ α

ω α α ω φ∗

Ψ Ψ = − Ψ − Ψ

= − − =

h

hh

( )22 �

2 2

? �( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2

( 1)2

xmp a a

m

ωξ α ξ α ξ α ξ α

ω α α α α α α∗ ∗ ∗

Ψ Ψ = − Ψ − Ψ

⎡ ⎤= − + − + −⎣ ⎦

h

h

2);0,(ˆ);0,();0,(ˆ);0,( 222 hωαξαξαξαξ mppp xxx =ΨΨ−ΨΨ=Δ

2h

=Δ⋅Δ xpx

Quantum Harmonic Oscillator - National Chiao Tung Universityocw.nctu.edu.tw/course/physics/quantummechanics_lecturenotes/chapter05.pdf · Quantum Harmonic Oscillator. 2006 Quantum

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