2D Quantum Harmonic Oscillatorocw.nctu.edu.tw/course/physics/quantummechanics_lecture...we will...

2D Quantum Harmonic Oscillator

2006 Quantum Mechanics Prof. Y. F. Chen

Two-Dimensional Quantum Harmonic Oscillator

in ch5, Schrödinger constructed the coherent state of the 1D H.O. to

describe a classical particle with a wave packet whose center in the

time evolution follows the corresponding classical motion

the H.O. plays a significant role in demonstrating the concept of

quantum-classical correspondence ∵ it can be analytically solved in

both CM & QM

the Schrödinger coherent state of the 2D H.O. is a nonspreading wave

packet with its center moving along the classical trajectories

we will start from the time-dep. Schrödinger coherent state for 2D H.O.

to extract the stationary coherent states that are localized on the

corresponding classical trajectories


2D Quantum Harmonic Oscillator2D Quantum Harmonic Oscillator

the Hamiltonian for the isotropic 2D H.O. in Cartesian coordinate：

the time-indep Schrödinger eq. is：

is separable：

→


Eigenstates of the 2D Isotropic Harmonic Oscillator2D Quantum Harmonic Oscillator

)(21

2222

22

yxmm

ppH yx ++

+= ω

),(),()(21

2222

2

2

2

22

yxEyxyxmyxm

ψψω =⎥⎦⎤

⎢⎣⎡ ++⎟

⎠⎞

⎜⎝⎛

∂∂

+∂∂

−h

),( yxψ )()(),( yxyx ΥΧ=ψ

Eymyd

dmy

xmxd

dmx

=⎟⎠⎞

⎜⎝⎛ +−

Υ+⎟

⎠⎞

⎜⎝⎛ +−

Χ22

2

2222

2

22

21

2)(1

21

2)(1 ωω hh

consequently, we have obtained 2 differential eq. for the 1D H.O.：

where

the eigenfunction and the eigenvalue of the 2D isotropic H.O. are given

by

where &



)()(21

222

2

22

xExxmxd

dm

x Χ=Χ⎟⎠⎞

⎜⎝⎛ +− ωh

)()(21

222

2

22

yEyymyd

dm

y Υ=Υ⎟⎠⎞

⎜⎝⎛ +− ωh

EEE yx =+

( ) 2 21/ 2 ( ) / 2, ( , ) 2 ! ! ( ) ( )x yn mm n x y m x n ym n e H Hξ ξψ ξ ξ π ξ ξ− − ++= ⋅%

( ) ωh1, ++= nmE nm

xmx hωξ = ymy hωξ =

the eigenvalues of the 2D H.O. are the sum of the two 1D oscillator

eigenenergies & the eigenfunctions are the product of two 1D

eigenfunctions

It can be found that the eigenstates in

do not reveal the characteristics of classical elliptical trajectories even in

the correspondence limit of large quantum number



( ) )()(!!2),(~ 2/)(2/1,22

ynxmmn

yxnm HHenm yx ξξπξξψξξ +−−+ ⋅=



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

(2,0) (0,2) (2,2) (5,5)

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

(2,0) (0,2) (2,2) (5,5)

Figure 7.1 Probability density patterns of eigenstates for the 2D isotropic harmonic oscillator

It is clear that the center of the wave packet follows the motion of a classical 2D

isotropic harmonic oscillator, i.e.,

The Schrödinger coherent state for the 2D isotropic harmonic oscillator is a

product of two infinite series. The method of the triangular partial sums is used

to make precise sense out of the product of two infinite series.

Mathematically, the notion of triangular partial sums is called the Cauchy product

of the double infinite series

Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator

2 cos( ) ; 2 cos( )x x x y y yt tξ α ω φ ξ α ω φ= − = −

With the representation of the Cauchy product, the terms can be arranged

diagonally by grouping together those terms for which has a fixed value:


2 2

2 2

( ) / 2 ( 1),

0 0

( ) / 2 ( 1),

0 0

( ) ( )( , , ) ( , )

! !

( ) ( ) ( , )

! ( ) !

yxx y

yxx y

ii m nx y i m n t

x y m n x yn m

ii K N KNx y i N t

K N K x yN K

e et e e

m n

e ee e

K N K

e

φφα α ω

φφα α ω

α αξ ξ ψ ξ ξ

α αψ ξ ξ

∞ ∞− + − + +

= =

−∞− + − +

−= =

−

Ψ =

=−

=

∑ ∑

∑ ∑

2 2( ) / 2 ( 1)

0

( ),

0

( )!

! ( , )

! ( ) !

y

x y

x y

i Nyi N t

N

KN

ixK N K x y

K y

ee

N

Ne

K N K

φα α ω

φ φ

α

α ψ ξ ξα

∞+ − +

=

−−

=

⎛⎜⎜⎝

⎞⎡ ⎤⎟× ⎢ ⎥⎟− ⎢ ⎥⎣ ⎦ ⎠

∑

∑

After some algebra,

The wave function above represents a type of normalized stationary coherent

state.


tNiyxN

NNyx eACt

ωφξξξξ )1(0

),;,(),,( +−∞

=

Φ=Ψ ∑ ( )!

1 22/)( 22

NeA

eCNi

yN

y

yx

φαα α+= +−

( ) ∑= −⎟⎠⎞

⎜⎝⎛

+=Φ

N

KyxKNK

KiNyxN AeK

N

AA

0,

2/1

2),(~)(

1

1),;,( ξξψφξξ φ yxy

xA φφφαα

−== ,


A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

Figure 7.2 Wave patterns of the stationary coherent states

for N=32 with different values of the parameters A and ψ.

2),;,( φξξ AyxNΦ


It can be seen that the coherent states correspond to the

elliptic stationary states.

The superposition of two elliptic states with a phase factor ψ

in the

opposite sign can form a standing wave pattern:

Next figure shows the standing wave patterns corresponding to the elliptic

states shown in figure above.

),;,( φξξ AyxNΦ

),;,(),;,( φξξφξξ −Φ±Φ AA yxNyxN


A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

A=1, φ = π /4 A=1, φ = π /3 A=1, φ = π /2

A=0.5, φ = π /2 A=1.5, φ = π /2 A=2.5, φ = π /2

Figure 7.3 Standing wave patterns corresponding to the elliptic states shown in figure 7.2.


manifestly reveals the relationship

between the Schrödinger coherent state and the stationary coherent state.

represents the probability of finding the system in the elliptic stationary

state with order N.

The probability distribution is identical to the Poisson distribution with the

mean value of

tNiyxN

NNyx eACt

ωφξξξξ )1(0

),;,(),,( +−∞

=

Φ=Ψ ∑

2NC

( ) )(222 22!

yxeN

CN

yxN

αααα +−+=

22yxN αα +>=

Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator

angular momentum of a classical particle is a vector quantity,

Angular momentum is the property of a system that describes the tendency

of an object spinning about the point r = 0 to remain spinning, classically.

For the motion of a classical 2D isotropic harmonic oscillator, the angular

momentum about the z-axis can be found to be independent of time:

prL ×=

t

mty

tm

tx

yy

xx

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=

)cos(||2)(

)cos(||2)(

φωαω

φωαωh

h

⎪⎪⎩

⎪⎪⎨

⎧

−−==

−−==

)sin(||2)()(

)sin(||2)()(

yyy

xxx

tmtdtydmtp

tmtdtxdmtp

φωαω

φωαω

h

h

)sin(||||2)()()()( yxyxxy tptytptx φφαα −−=− h


In quantum mechanics, the angular momentum is associated with the

operator , that is defined as

For 2D motion the angular momentum operator about the z-axis is

The expectation value of the angular momentum for the stationary coherent

state and time-dependent wave packet state which are shown below :

L̂ prL ˆˆˆ ×=

xyz pypxL ˆˆˆˆˆ −=

( ) ∑= −⎟⎠⎞

⎜⎝⎛

+=Φ

N

KyxKNK

KiNyxN AeK

N

AA

0,

2/1

2),(~)(

1

1),;,( ξξψφξξ φ

tNiyxN

NNyx eACt

ωφξξξξ )1(0

),;,(),,( +−∞

=

Φ=Ψ ∑


The position and momentum operators for the harmonic oscillator can be in

terms of the creation and annihilation operators.

( ) ( ) ( ) ( )

( )

? ?

?

ˆ ? ?

1 ? ? ? ?2

? ?

z y x

x x y y y y x x

x y x y

L x p y p

i a a a a a a a a

i a a a a

= −

⎡ ⎤= + − − + −⎣ ⎦

= −

h

h


The properties of the creation and annihilation operators :

( )∑=

+−−+−⎟⎠⎞

⎜⎝⎛

+=

Φ

N

KyxKNK

KiN

yxNyx

AeKNKKN

A

Aaa

11,1

2/1

2/2

†

),(~1)1(

1

),;,(ˆˆ

ξξψ

φξξ

φ

( )∑−

=−−+−+⎟

⎠⎞

⎜⎝⎛

+=

Φ

1

01,1

2/1

2/2

†

),(~1)1(

1

),;,(ˆˆN

KyxKNK

KiN

yxNxy

AeKNKKN

A

Aaa

ξξψ

φξξ

φ


With the orthonormal property of the eignestates :

( )∑

∑

=

−−

−

=

−+

⎟⎠⎞

⎜⎝⎛

+=

−+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

++=

ΦΦ

N

K

iKN

N

K

iKN

yxNyxyxN

AeAKKN

A

eAKNKKN

KN

A

AaaA

1

)1(22

1

0

122/12/1

2

†

)1(1

11)1(

1

),;,(|ˆˆ|),;,(

φ

φ

φξξφξξ

( )∑

∑

=

−

=

−

⎟⎠⎞

⎜⎝⎛

+=

+−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

−+=

ΦΦ

N

K

iKN

N

K

iKN

yxNyxyxN

AeAKKN

A

eAKNKKN

KN

A

AaaA

1

)1(22

1

122/12/1

2

†

)1(1

11)1(

1

),;,(|ˆˆ|),;,(

φ

φ

φξξφξξ


Using the property

We can obtain and

( ) ( ) ∑∑=

−−

=⎟⎠⎞

⎜⎝⎛=+⇒⎟

⎠⎞

⎜⎝⎛

∂∂

=+∂∂ N

K

KNN

K

KN xKKN

xNxKN

xx

x 111

0

11

)1()1(1

21

)1(22 A

NAKKN

A

N

K

KN +

=⎟⎠⎞

⎜⎝⎛

+ ∑=−

( )

( )

?

2( 1)2 2

1

ˆ( , ; , ) | | ( , ; , )

? ?( , ; , ) | | ( , ; , )

2 sin(1 ) 1

N x y z N x y

N x y x y x y N x y

NK i i

NK

A L A

A i a a a a A

Ni AK A Ae Ae NKA A

φ φ

ξ ξ φ ξ ξ φ

ξ ξ φ ξ ξ φ

φ− −=

Φ Φ

= Φ − Φ

⎛ ⎞= − = −⎜ ⎟+ +⎝ ⎠

∑

h

hh

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits


The time-independent Schrödinger equation for a 2D harmonic oscillator

with commensurate frequencies can generally given by

is the common factor of the frequencies by and , and p and q are relative prime integers

The eigenfunction and the eigenvalue of the 2D harmonic oscillator with

commensurate frequencies are given by

),(),()(21

22222

2

2

2

22

yxEyxyxmyxm yx

ψψωω =⎥⎦⎤

⎢⎣⎡ ++⎟

⎠⎞

⎜⎝⎛

∂∂

+∂∂

−h

ωω qx = ωω py =

xω yωω

( ) )()(!!2),(~ 2/)(2/1,22

ynxmmn

yxnm HHenm yx ξξπξξψξξ +−−+ ⋅=

yxnm nmE ωω hh ⎟⎠⎞⎜

⎝⎛ ++⎟

⎠⎞⎜

⎝⎛ +=

21

21

, xm xx hωξ = ym yy hωξ =



The eigenfunction is separable, so the corresponding Schrödinger coherent

state can be expressed as the product of two 1D coherent states:

∑ ∑

∑

∑

∞

=

∞

=

+++−+−

∞

=

+−−−

∞

=

+−−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛×

⎟⎟⎠

⎞⎜⎜⎝

⎛=Ψ

0 0

)2/2/(,

2/)(

0

)2/1(2/2/

0

)2/1(2/2/

),(~!!

)()(

)(!2

1!

)(

)(!2

1!)(),,(

22

22

22

n m

tpqpnqmiyxnm

niy

mix

n

tpniyn

n

niy

m

tqmixm

m

nix

yx

eenm

ee

eeHn

en

e

eeHm

emet

yx

yx

yy

y

xxx

ωααφφ

ωξαφ

ωξαφ

ξξψαα

ξπ

α

ξπ

αξξ



It is clear that the center of the wave packet follows the motion of a classical

2D isotropic harmonic oscillator, i.e.,

The set of states with indices in last page can be divided into subsets

characterized by a pair of indices given by and

Schrödinger coherent state can be rewritten as

)cos(2;)cos(2 yyyxxx tptq φωαξφωαξ −=−=

),( nm

),( yx uu ( )pum x mod≡ ( )qun y mod≡

)

2 21 1 ( ) / 2

0 0 0 0

[ ( ) ( 1/ 2) ( 1/ 2)],

( ) ( )( , , )

( ) ! ( ) !

( , )

y y yx x xx y

y x y x

x y x y

x x y y

i qN ui pN uq px y

x yu u N N x x y y

i pq N N q u p u tpN u qN u x y

e et e

pN u qN u

e

φφα α

ω

α αξ ξ

ψ ξ ξ

++− − ∞ ∞− +

= = = =

− + + + + ++ +

⎛⎜Ψ =⎜ + +⎝

×

∑ ∑ ∑ ∑

%



The 2D Schrödinger coherent state is divided into a product of two infinite

series and two finite series

With the representation of the Cauchy product, the terms

can be arranged diagonally by grouping together those terms for which

:

),(~ , yxuqNupN yyxx ξξψ ++

NNN yx =+

)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−+×

=

×

⎜⎜⎝

⎛

+−+=Ψ

+−+=

−

−

=

−

=

∞

=

++++−++−

++++−+−+

−

=

−

=

∞

= =

+−+−+

∑

∑ ∑∑

∑ ∑∑∑

),(~!])([!)(

][)/(

)()(

),(~!])([!)(

)()(),,(

)(,0

)(

1

0

1

0 0

)]2/1()2/1([2/)(

)]2/1()2/1([)(,

1

0

1

0 0 0

2/)()(

22

22

yxuKNqupK

N

K yx

KqpiKqy

px

q

u

p

u N

tupuqpqNiuqNiy

uix

tupuqpqNiyxuKNqupK

q

u

p

u N

N

K yx

uKNqiy

upKix

yx

yx

yx

y x

yxyyxxyx

yx

yx

y x

yx

yyxx

uKNqupK

e

eeee

e

euKNqupK

eet

ξξψαα

αα

ξξψ

ααξξ

φφ

ωφφαα

ω

ααφφ



These stationary coherent states are physically expected to be associated

with the Lissajous trajectories.

The minor indices ux and uy essentially do not affect the characteristics of

the stationary states.

Including the normalization condition, the stationary coherent states in

Cartesian coordinates are given by

),(~!])([!)(

][

!])([!)(),;,(

)(,0

2/1

0

2

,,

yxuKNqupK

N

K y

Ki

N

K y

K

yxuuN

yx

yx

uKNqpKeA

uKNqpKAA

ξξψ

φξξ

φ

+−+=

−

=

∑

∑

+−×

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−⋅=Φ

yxqy

px qpA φφφ

αα

−== ,)()(



The stationary coherent states associated with the Lissajous trajectories are

the superposition of degenerate eigenstates with the relative amplitude

factor A and phase factor .

The relative amplitude factor A and phase factor in the stationary

coherent states are explicitly related to the classical

variables

the eigenenergies of the stationary coherent states

are found to be

φ

φ

),;,(,, φξξ AyxuuN yxΦ

( )yxyx φφαα ,,,),;,(,, φξξ AyxuuN yxΦ

[ ] ωh)2/1()2/1(,, ++++= yxuuN upuqpqNE yx



Next three figures depict the comparison between the quantum wave

patterns and the corresponding classical periodic

orbits for to be , , and , respectively.

Three different phase factors, , , and , are displayed

in each figure.

The behavior of the quantum wave patterns in all cases can be found to be

in precise agreement with the classical Lissajous figures.

20,0, ),;,( φξξ AyxNΦ

qp : 1:2 3:42:3

0=φ πφ 3.0= πφ 6.0=



(a) (b) (c)

(a’) (b’) (c’)

(a) (b) (c)

(a’) (b’) (c’)



(a) (b) (c)

(a’) (b’) (c’)

(a) (b) (c)

(a’) (b’) (c’)

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2D Quantum Harmonic Oscillatorocw.nctu.edu.tw/course/physics/quantummechanics_lecture...we will...

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