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
2006 Quantum Mechanics Prof. Y. F. Chen
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:
→
2006 Quantum Mechanics Prof. Y. F. Chen
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 &
2006 Quantum Mechanics Prof. Y. F. Chen
Eigenstates of the 2D Isotropic Harmonic Oscillator2D Quantum Harmonic Oscillator
)()(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
2006 Quantum Mechanics Prof. Y. F. Chen
Eigenstates of the 2D Isotropic Harmonic Oscillator2D Quantum Harmonic Oscillator
( ) )()(!!2),(~ 2/)(2/1,22
ynxmmn
yxnm HHenm yx ξξπξξψξξ +−−+ ⋅=
2006 Quantum Mechanics Prof. Y. F. Chen
Eigenstates of the 2D Isotropic Harmonic Oscillator2D Quantum Harmonic Oscillator
(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:
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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.
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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
−== ,
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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.
Stationary Coherent States of the 2D Isotropic H.O.2D Quantum Harmonic Oscillator
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
Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator
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 ∑
Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator
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
Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator
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
Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator
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
Angular Momentum in 2D Confined Systems2D Quantum Harmonic Oscillator
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
2D Quantum Harmonic Oscillator
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 =
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
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
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum 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 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 =⎜ + +⎝
×
∑ ∑ ∑ ∑
%
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
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
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
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
−== ,)()(
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
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
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
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=
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
(a) (b) (c)
(a’) (b’) (c’)
(a) (b) (c)
(a’) (b’) (c’)
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
(a) (b) (c)
(a’) (b’) (c’)
(a) (b) (c)
(a’) (b’) (c’)
Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits
2D Quantum Harmonic Oscillator
(a) (b) (c)
(a’) (b’) (c’)
(a) (b) (c)
(a’) (b’) (c’)
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