2D Quantum Harmonic Oscillator

Two-Dimensional Quantum Harmonic Oscillator

2006 Quantum Mechanics Prof. Y. F. Chen 2D Quantum Harmonic Oscillator 2D 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 Oscillator Eigenstates of the 2D Isotropic Harmonic Oscillator

the Hamiltonian for the isotropic 2D H.O. in Cartesian coordinate： + pp 22 1 H = yx + ω + yxm 222 )( 2m 2 the time-indep Schrödinger eq. is： ⎡ ⎛ ∂22 ∂2 ⎞ 1 ⎤ − h + + ω + 222 ψ = ψ yxEyxyxm ),(),()( ⎢ ⎜ 2 2 ⎟ ⎥ ⎣ 2 ⎝ ∂ ∂ yxm ⎠ 2 ⎦

ψ yx ),( is separable：ψ = Χ Υ yxyx )()(),(

1 ⎛ d 22 1 ⎞ 1 ⎛ d 22 1 ⎞ ⎜ h +− ω xm 22 ⎟ + ⎜ h +− ω 22 ⎟ = Eym → Χ ⎝ 2)( mx xd 2 2 ⎠ Υ ⎝ 2)( my yd 2 2 ⎠

2006 Quantum Mechanics Prof. Y. F. Chen 2D Quantum Harmonic Oscillator Eigenstates of the 2D Isotropic Harmonic Oscillator

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

⎛ d 22 1 ⎞ ⎜ h +− ω 22 ⎟ x Χ=Χ xExxm )()( ⎝ 2m xd 2 2 ⎠ ⎛ d 22 1 ⎞ ⎜ h +− ω 22 ⎟ y Υ=Υ yEyym )()( ⎝ 2m yd 2 2 ⎠

where E + E yx = E

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

−1/2 22 nm+ −+()/2ξξxy by ψ% mn, (,ξξ x y )=⋅( 2mn ! !π) e Hm()()ξ x ξ H n y

,nm ()+= nmE + 1 hω

where x = ωξ h xm & y = ωξh ym

2006 Quantum Mechanics Prof. Y. F. Chen 2D Quantum Harmonic Oscillator Eigenstates of the 2D Isotropic Harmonic Oscillator

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

− 2/1 22 ~ +mn +− ξξyx 2/)( , ξξψyxnm = ( !!2),( ⋅π ) HHenm ξξynxm )()(

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 2D Quantum Harmonic Oscillator Eigenstates of the 2D Isotropic Harmonic Oscillator

(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

2006 Quantum Mechanics Prof. Y. F. Chen 2D Quantum Harmonic Oscillator Stationary Coherent States of the 2D Isotropic H.O.

It is clear that the center of the wave packet follows the motion of a classical 2D isotropic harmonic oscillator, i.e.,

ξx =−=−2cos();2cos()αωφξαωφxxttyyy

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

With the representation of the Cauchy product, the terms can be arranged diagonally by grouping together those terms for which has a fixed value:

iφ ∞∞ iφx mny ()()ααee22 xy−+()/2ααxy −++im(1) nω t Ψ=(,ξξxy ,)tee∑∑ ψmnxy ξξ, (, ) nm==00 mn!!

iφ ∞ N iφx KNy −K ()()ααee 22 xy −+()/2ααxy −+iN(1)ω t = ∑∑ eeψξξKN, − K(,) x y NK==00 KNK!(− )!

∞ iφy N ⎛ 22 ()α e = ⎜ e−()/2ααxy+ e−+iN(1)ω t y ⎜ ∑ ⎝ N =0 N ! K N ⎞ N ! ⎡⎤α i() x φφxy− ⎟ × ∑ ⎢⎥e ψξξKN, − K(,) x y K =0 KNK!(− ) ! α ⎟ ⎣⎦⎢⎥y ⎠ 2D Quantum Harmonic Oscillator Stationary Coherent States of the 2D Isotropic H.O.

After some algebra,

∞ N 2 iφy +− )1( ω tNi 22 ( 1+ α eA ) Ψ ξξ ),,( Ct Φ= φξξ),;,( eA +− ααyx 2/)( y yx ∑ N yxN N = eC N =0 N !

N 2/1 1 ⎛ N ⎞ α x Φ A φξξ),;,( = Ae φ )( Ki ~ ξξψ),( A = , −= φφφ yxN N ∑ ⎜ ⎟ , − yxKNK yx 2 K ( 1+ A ) K =0 ⎝ ⎠ α y

The wave function above represents a type of normalized stationary coherent state. 2D Quantum Harmonic Oscillator Stationary Coherent States of the 2D Isotropic H.O.

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

2 Figure 7.2 Wave patterns of the stationary coherent states Φ yxN A φξξ),;,(

for N=32 with different values of the parameters A and ψ. 2D Quantum Harmonic Oscillator Stationary Coherent States of the 2D Isotropic H.O.

It can be seen that the coherent states Φ ξ ξ yxN A φ ),;,( 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:

Φ ξ ξ yxN A φ ± Φ ξ ξ yxN A −φ),;,(),;,(

Next figure shows the standing wave patterns corresponding to the elliptic states shown in figure above. 2D Quantum Harmonic Oscillator Stationary Coherent States of the 2D Isotropic H.O.

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

∞ +− )1( ω tNi Ψ ξξyx ),,( ∑ Ct N Φ= yxN φξξ),;,( eA manifestly reveals the relationship N =0 between the Schrödinger coherent state and the stationary coherent state.

2 CN represents the probability of finding the system in the elliptic stationary state with order N.

22 N ()+ αα 22 C 2 = yx e +− ααyx )( N N !

The probability distribution is identical to the Poisson distribution with the

22 mean value of N +>=< ααyx 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

angular momentum of a classical particle is a vector quantity, = × prL

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:

⎧ txd )( h ⎧ )( mtp −== m t − φωαω)sin(||2 ⎪ tx = x cos(||2)( t − φωαx ) x h x x ⎪ mω ⎪ td ⎨ ⎨ ⎪ h ⎪ tyd )( ty = y cos(||2)( t − φωαy ) y )( = mtp −= m h y t − φωαωy )sin(||2 ⎩⎪ mω ⎩⎪ td

y − x tptytptx −= h yx − φφααyx )sin(||||2)()()()( 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

In quantum mechanics, the angular momentum is associated with the

operator L ˆ , that is defined as ˆ ×= prL ˆˆ

For 2D motion the angular momentum operator about the z-axis is ˆ z y −= pypxL ˆˆˆˆ x The expectation value of the angular momentum for the stationary coherent state and time-dependent wave packet state which are shown below :

2/1 1 N ⎛ N ⎞ Φ A φξξ),;,( = Ae φ )( Ki ~ ξξψ),( yxN N ∑ ⎜ ⎟ , − yxKNK ( 1+ A2 ) K =0 ⎝ K ⎠

∞ +− )1( ω tNi Ψ ξξyx ),,( ∑ Ct N Φ= yxN φξξ),;,( eA N =0 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

The position and momentum operators for the harmonic oscillator can be in terms of the creation and annihilation operators.

ˆ Lxpypzyx=−??

1 =+−−+−iaaaaaaaa⎡⎤()()()()???? ?? h 2 ⎣⎦xx yy yy xx

? =−iaaaah ()??xy x y 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

The properties of the creation and annihilation operators :

† aa ˆˆ Φ yxNyx A φξξ),;,(

N 2/1 1 ⎛ N ⎞ K ()iφ ~ = N 2/2 ∑ ⎜ ⎟ +− 1 AeKNK +−− 1,1 ξξψyxKNK ),( + A )1( K=1 ⎝ K ⎠

† aa ˆˆ Φ yxNxy A φξξ),;,(

N −1 2/1 1 ⎛ N ⎞ K ()iφ ~ = N 2/2 ∑ ⎜ ⎟ 1 −+ AeKNK −−+ 1,1 ξξψyxKNK ),( + A )1( K=0 ⎝ K ⎠ 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

With the orthonormal property of the eignestates :

† Φ yxN φξξ|),;,( aaA ˆˆ Φ yxNyx A φξξ),;,(|

N 2/12/1 1 ⎛ N ⎞ ⎛ N ⎞ −12 iK φ = 2 N ∑ ⎜ ⎟ ⎜ ⎟ +− 1 eAKNK + A )1( K =1 ⎝ K −1⎠ ⎝ K ⎠

1 N ⎛ N ⎞ K − )1(2 ()iφ = 2 N ∑ ⎜ ⎟ AeAK + A )1( K =1 ⎝ K ⎠

† Φ yxN φξξ|),;,( aaA ˆˆ Φ yxNyx A φξξ),;,(|

N −1 2/12/1 1 ⎛ N ⎞ ⎛ N ⎞ 12 −+ iK φ = 2 N ∑ ⎜ ⎟ ⎜ ⎟ 1 −+ eAKNK + A )1( K=0 ⎝ K +1⎠ ⎝ K ⎠

1 N ⎛ N ⎞ K )1(2 ()−− iφ = 2 N ∑ ⎜ ⎟ AeAK + A )1( K=1 ⎝ K ⎠ 2D Quantum Harmonic Oscillator Angular Momentum in 2D Confined Systems

Using the property

N N ∂ N ∂ ⎛ N ⎞ N −1 ⎛ N ⎞ ()1 x =+ ∑ ⎜ ⎟ K ()1 xNx =+⇒ ∑ ⎜ ⎟ xK K −1 ∂x ∂x K =0 ⎝ K ⎠ K =1 ⎝ K ⎠

N 1 ⎛ N ⎞ K− )1(2 N We can obtain 2 N ∑ ⎜ ⎟ AK = 2 and + A )1( K=1 ⎝ K ⎠ + A )1(

ˆ ΦΦNxy(,;,)||ξξ φ ξξAL φ z Nxy (,;,) A

? =ΦNxy(,;,)|ξξ φAiaaaah()??xy− xy | Φ Nxy (,;,)ξξ φ A iAN ⎛⎞N h K A2(Ki−− 1) Aeφφ Ae i2sin N =−2 N ∑ ⎜⎟ ()=−h 2 φ (1++AA )K =1 ⎝⎠K 1 2D Quantum Harmonic Oscillator

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

⎡ ⎛ ∂22 ∂2 ⎞ 1 ⎤ − h + + ( + 2222 ψωω= ψ yxEyxyxm ),(),() ⎢ ⎜ 2 2 ⎟ x y ⎥ ⎣ 2 ⎝ ∂ ∂ yxm ⎠ 2 ⎦

ωx = qω ω y = pω

ω is the common factor of the frequencies by ω x and ω y , and p and q are relative prime integers The eigenfunction and the eigenvalue of the 2D harmonic oscillator with commensurate frequencies are given by − 2/1 22 ~ +mn +− ξξyx 2/)( , ξξψyxnm = ( !!2),( ⋅π ) HHenm ξξynxm )()(

⎛ 1 ⎞ ⎛ 1 ⎞ ,nm ⎜mE += ⎟ hωx ⎜n ++ ⎟ hωy x = ωξx xm y = ωξy ym ⎝ 2 ⎠ ⎝ 2 ⎠ h h 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

The eigenfunction is separable, so the corresponding Schrödinger coherent state can be expressed as the product of two 1D coherent states:

∞ iφx n ⎛ α e )( 2 1 2 ⎞ ⎜ x −α x 2/ −ξ x 2/ +− )2/1( ω tqmi ⎟ Ψ ξξyx t),,( = e ξ xm )( eeH ⎜∑ m ⎟ ⎝ m=0 m! m!2 π ⎠ iφ ∞ y n ⎛ α e )( −α 2 2/ 1 −ξ 2 2/ ⎞ ⎜ y y y +− )2/1( ω tpni ⎟ × e ξ yn )( eeH ⎜∑ n ⎟ ⎝ n=0 n! n!2 π ⎠

iφ ∞ ∞ iφx m y n ααee )()( 22 x y +− ααyx 2/)( ~ ( +++− )2/2/ ω tpqpnqmi = ∑∑ e , ξξψyxnm ),( e nm=00= nm !! 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

It is clear that the center of the wave packet follows the motion of a classical 2D isotropic harmonic oscillator, i.e.,

cos(2 x = x cos(2 tq − x y = y cos(2;) tp −φωαξφωαξy )

The set of states with indices nm ),( in last page can be divided into subsets

uu ),( characterized by a pair of indices yx given by ≡ x (mod pum ) and ≡ y (mod qun )

Schrödinger coherent state can be rewritten as

iqNuφ + qp−−11∞∞ ipNuφxxx+ yyy ⎛ ()()ααee22 ⎜ xy−+()/2ααxy Ψ=(,ξξxy ,)te∑∑∑∑ ⎜ uuNN====0000()!()!pN++ u qN u ⎝ yxyx xx yy

×ψξξ(, )e−+++++ipqN[(xy N )( qu x 1/2)( pu y 1/2)]ω t % pNxx++ u, qN yy u x y ) 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

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

ψ~ ξ ξ ),( With the representation of the Cauchy product, the terms , ++ yyxx yxuqNupN can be arranged diagonally by grouping together those terms for which

yx =+ NNN :

φ )( +− uKNqi q−1 p−1 ∞ N φx +upKi x y y ⎛ α e α e )()( 22 Ψ ξξt),,( = ⎜ x y e +− ααyx 2/)( yx ⎜ ∑∑∑∑ ⎝ u yx=0 uN=000==K + x +− uKNqupK y !])([!)( ×ψ~ ξξ),( e x upuqpqNi y ++++− )]2/1()2/1([ ω t ) x )(, +−+ y yxuKNqupK

q−1 p−1 ∞ 22 +− ααyx 2/)( φ ui xx φy y x upuqpqNiuqNi y ++++−+ )]2/1()2/1([ ω t = ∑∑∑e x ααy )()( eee u yx=0 uN=00= −qpi φφ)( N p Kq yx K ⎪⎧ x ααy e ][)/( ⎪⎫ × ψ~ ξξ),( ⎨∑ x )(, +−+ y yxuKNqupK ⎬ ⎩⎪K =0 + x +− uKNqupK y !])([!)( ⎭⎪ 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

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

− 2/1 ⎛ N A2K ⎞ Φ A φξξ),;,( = ⎜ ⎟ ,, yx yxuuN ⎜ ∑ ⎟ ⎝ K =0 +−⋅ uKNqpK y !])([!)( ⎠ N eA φ ][ Ki × ψ~ ξξ),( ∑ x )(, +−+ y yxuKNqupK K =0 +− uKNqpK y !])([!)(

α )( p A = x , −= qp φφφ q yx α y )( 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

The stationary coherent states associated with the Lissajous trajectories are the superposition of degenerate eigenstates with the relative amplitude factor A and phase factor φ .

The relative amplitude factor A and phase factor φ in the stationary coherent states Φ ξ ξ A φ ),;,( are explicitly related to the classical ,, yx yxuuN

variables ()α α φ ,,, φ yxyx the eigenenergies of the stationary coherent states Φ ξ ξ A φ),;,( ,, yx yxuuN are found to be

E = []+ uqpqN + + up + )2/1()2/1( ω ,, uuN yx x y h 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

Next three figures depict the comparison between the quantum wave 2 patterns Φ 0,0, yxN A φξξ ),;,( and the corresponding classical periodic orbits for p : q to be 1:2 , 2:3 , and 3:4 , respectively.

Three different phase factors, φ = 0 , φ = 3.0 π , and φ = 6.0 π , 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. 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

(a) (b) (c)

(a’) (b’) (c’) 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

(a) (b) (c)

(a’) (b’) (c’) 2D Quantum Harmonic Oscillator

Quantum Stationary Coherent States for Classical Lissajous Periodic Orbits

(a) (b) (c)

(a’) (b’) (c’)