2-2

CompConj[A_] := A /.Complex[0,n_]->-Complex[0,n]

Quantum Motion of a Particle in a Box

Nicholas Wheeler
Reed College Physics Department
18 August 2008

Introduction

It is by way of preparation for discussion of "quantum Liossajous figures" and "quantum vorticity" that I look here to aspects—specifically, dynamics aspects—of the simplest and most familiar of all quantum mechanical systems: the "particle-in-a-box" problem. I work first in one dimension, then by straightforward generalization in two dimensions.

Two-dimensional theory

We redirect our attention now to properties of the solutions of

-((ℏ^2)/(2m))((∂_(x,x))+(∂_(y,y)))ψ[x,y,t]=i ℏ (∂_t) ψ[x,y,t]

that conform to the boundary conditions

ψ[0,0,t]=ψ[a,0,t]=ψ[0,a,t]=ψ[a,a,t]=0    :    all  t

If we adopt units in which

ℏ=2m=a=1

we have

-((∂_(x,x))+(∂_(y,y)))ψ[x,t]=i  (∂_t) ψ[x,t]
ψ[0,t]=ψ[1,t]    :    all  t

which is easier to work with. If we assume the wave function to have separated structure

ψ[x,y,t]=(Subscript[Ψ, 1])[x]*(Subscript[Ψ, 2])[y]*(e^(-i ω t))

and are led immediately to the orthonormal eigenfunctions

Ψ[x_,y_,m_,n_]:=2 Sin[m π x]Sin[n π y]

Plot3D[Ψ[x,y,2,3],{x,0,1},{y,0,1},LabelStyle->Directive[Bold], PlotLabel->"The {2, 3} Eigenstate" ]
Plot3D[((Ψ[x,y,2,3])^2),{x,0,1},{y,0,1},LabelStyle->Directive[Bold], PlotLabel->"The {2, 3} Probability Distribution"]

From

-(∂_(x,x))Ψ[x,y,m,n]-(∂_(y,y))Ψ[x,y,m,n]==((m^2)+(n^2)) (π^2)Ψ[x,y,m,n]//Simplify

we are led to define the "eigenfrequencies"

ω[m_,n_]:=((m^2)+(n^2))(π^2)

and to construct the "buzz factors"

Clear[BuzzFactor]

BuzzFactor[m_,n_,t_]:=(E^(-I ω[m,n] t))

The time-dependent eigenfunctions are given now by

ψ[x_,y_,m_,n_,t_]:=Ψ[x,y,m,n]*BuzzFactor[m,n,t]

which we verify do satisfy the time-dependent Schrøodinger equation:

-(∂_(x,x))ψ[x,y,m,n,t]-(∂_(y,y))ψ[x,y,m,n,t]==I (∂_t) ψ[x,y,m,n,t]//Simplify

The function ψ[x, y, m, n, t], since complex-valued, does not lend itself to graphic representation (people sometimes graph the amplitude, and use color to indicate the t-dependent phase), and there is no reason to graph the "moving probability amplitude," since it does not move! To see motion one must look to weighted linear combinations of eigenfunctions that buzz with distinct frequencies.

Motion of simple 2D superpositions

One must be careful to distinguish superpositions of the form

ϕ[x,y,t]=((∑_m)(Subscript[A, m]) ψ[x,m,t])*((∑_n)(Subscript[B, n]) ψ[x,n,t])

— which are assembled from products of sums of one-dimensional buzzing eigenfunctions

Ψ[x_,n_]:=(Sqrt[2]) Sin[n π x]
ψ[x_,n_,t_]:=Ψ[x,n]*(E^(-I (n^2)(π^2)t))

—from superpositions of the form

Φ[x,y,t]=(∑_m)(∑_n)(Subscript[C, mn]) ψ[x,y,m,n,t]

which are sums of products of one-dimensional buzzing eigenfunctions. The former give rise in the classical limit to Lissajous figures, and the latter to more complex kinds of quantum motion. I look to the alternatives in the order stated.

First approach to the quantum theory of Lissajous figures

The classical motion of a partticle that moves freely within a one-dimensional bounding box is shown in the figure which I now construct:

A=Table[{{2n-x0-v t,(((2n-1)-x0)/v)<t<((2n-x0)/v)},
{v t-2n+x0,((2n-x0)/v)<t<(((2n+1)-x0)/v)}},{n,0,50}];

Segments={Part[A,1,1],Part[A,1,2],Part[A,2,1],Part[A,2,2],
Part[A,3,1],Part[A,3,2],Part[A,4,1],Part[A,4,2],
Part[A,5,1],Part[A,5,2],Part[A,6,1],Part[A,6,2],
Part[A,7,1],Part[A,7,2],Part[A,8,1],Part[A,8,2],
Part[A,9,1],Part[A,9,2],Part[A,10,1],Part[A,10,2],
Part[A,11,1],Part[A,11,2],Part[A,12,1],Part[A,12,2],
Part[A,13,1],Part[A,13,2],Part[A,14,1],Part[A,14,2],
Part[A,15,1],Part[A,15,2],Part[A,16,1],Part[A,16,2],
Part[A,17,1],Part[A,17,2],Part[A,18,1],Part[A,18,2],
Part[A,19,1],Part[A,19,2],Part[A,20,1],Part[A,20,2],
Part[A,21,1],Part[A,21,2],Part[A,22,1],Part[A,22,2],
Part[A,23,1],Part[A,23,2],Part[A,24,1],Part[A,24,2],
Part[A,25,1],Part[A,25,2],Part[A,26,1],Part[A,26,2],
Part[A,27,1],Part[A,27,2],Part[A,28,1],Part[A,28,2],
Part[A,29,1],Part[A,29,2],Part[A,30,1],Part[A,30,2],
Part[A,31,1],Part[A,31,2],Part[A,32,1],Part[A,32,2],
Part[A,33,1],Part[A,33,2],Part[A,34,1],Part[A,34,2],
Part[A,35,1],Part[A,35,2],Part[A,36,1],Part[A,36,2],
Part[A,37,1],Part[A,37,2],Part[A,38,1],Part[A,38,2],Part[A,39,1],Part[A,39,2],Part[A,40,1],Part[A,40,2],
Part[A,41,1],Part[A,41,2],Part[A,42,1],Part[A,42,2],
Part[A,43,1],Part[A,43,2],Part[A,44,1],Part[A,44,2],
Part[A,45,1],Part[A,45,2],Part[A,46,1],Part[A,46,2],Part[A,47,1],Part[A,47,2],Part[A,48,1],Part[A,48,2],
Part[A,49,1],Part[A,49,2],Part[A,50,1],Part[A,50,2]};

x[t_]:=Piecewise[Segments]

That function serves to describe the motion of such a particle for times

Assigning particular values to the speed and initial position

X[t_]:=x[t]/.{x0->0.4, v->1}

we obtain the figure

Plot[Evaluate[X[t]],{t,0,21},PlotRange->{0,1.05}, GridLines->{{},{1}}, Ticks->{{5,10,15,20},{1}}, AxesLabel->{"TIME","POSITION"}]

A description of the motion of a free particle trapped within a 2-dimensional box is obtained by forming the tensor product of the preceding construction:

X[t_] := x[t] /. {x0 -> .4, v -> 1}
Y[t_] := x[t] /. {x0 -> 0, v -> 3}
ParametricPlot[Evaluate[{X[t],Y[t]}], {t,0,20},GridLines->{{1},{1}}, Ticks->False, AxesLabel->{x,y}, LabelStyle->Large]

Here is a slightly de-tuned variant of the preceding figure:

X[t_] := x[t] /. {x0 -> .4, v -> 1}
Y[t_] := x[t] /. {x0 -> 0, v -> 3.01}
ParametricPlot[Evaluate[{X[t],Y[t]}], {t,0,10},GridLines->{{1},{1}}, Ticks->False, AxesLabel->{x,y}, LabelStyle->Large]

The following commands permit one to construct unlimitedly many slightly de-tuned instances of the simplest kind of Lissajous figures—those in which right/left and up/down traversals occur with the same (or nearly the same) period:

X[t_] := x[t] /. {x0 ->RandomReal[], v ->1}
Y[t_] := x[t] /. {x0 ->RandomReal[], v ->1.005}
ParametricPlot[Evaluate[{X[t],Y[t]}], {t,0,30},GridLines->{{1},{1}}, Ticks->False, AxesLabel->{x,y}, LabelStyle->Large]

I turn now to the construction of a quantum state that executes a "soft" Lissajous figure. It proceeds from the assumption that x and y are independent random variables. From

Clear[x]

ϕx[x_,t_,α_]:=Cos[α](Sqrt[2])Sin[π 1 x](E^(-I (1^2)t))+Sin[α](Sqrt[2])Sin[π 2 x](E^(-I (2^2)t))
ϕy[y_,t_,β_]:=Cos[β](Sqrt[2])Sin[π 3 y](E^(-I (3^2)t))+Sin[β](Sqrt[2])Sin[π 4 y](E^(-I (4^2)t))

(where α and β are "mixing angles") construct

We construct the probability density associated with ϕx[x, t, α] and

Px[x_,t_,α_]:=(((ComplexExpand[Re[ϕx[x,t,α]]]))^2)+(((ComplexExpand[Im[ϕx[x,t,α]]]))^2)

TrigReduce[Px[x,t,α]]

and see that it has angular frequency

(Subscript[Ω, x])=3

Experimentation leads us to set α  = 0.64 (in order to get a good clear slosh); the moving distribution looks like this:

Animate[Plot[Px[x,t,0.64],{x,0,1}, PlotRange->{0,3}],{t,0,((2π)/3)}]

Similarly,

Py[y_,t_,β_]:=(((ComplexExpand[Re[ϕy[y,t,β]]]))^2)+(((ComplexExpand[Im[ϕy[y,t,β]]]))^2)

TrigReduce[Py[x,t,β]]

is seen to have angular frequency

(Subscript[Ω, y])=7

Experimentation leads us to set β = 3.77, producing a sloshing distribution that looks like this:

Animate[Plot[Py[y,t,3.77],{y,0,1}, PlotRange->{0,4}],{t,0,((2π)/7)}]

We assume the composite wavefunction to have the factored structure

Φ[x_,y_,t_]:=ϕx[x,t,0.64]*ϕy[y,t,3.77]

Φ[x,y,t]

which is, in effect, to assume that x and y are independent random variables.

We assemble the associated probability density

P[x_,y_,t_]:=(((ComplexExpand[Re[Φ[x,y,t]]]))^2)+(((ComplexExpand[Im[Φ[x,y,t]]]))^2)

This is seen to be quite a complicated expression

TrigReduce[P[x,y,t]]//Chop

Length[%]

in which the time variable enters (via cross terms—interference terms) with the following factors:

{3, 4, 7, 10}

of which the least common multiple is

LCM[3,4,7,10]

But  P[x, y, t]  is actually not so complicated as it seems. It is the product of two simple slosh functions that have already been shown to have angular frequencies 3 and 7, respectively:

PXTPYT=Expand[Px[x,t,0.64]*Py[y,t,3.77]];
PXYT=Expand[P[x,y,t]];

PXYT-PXTPYT//Chop

The 4 and 10 arise as artifacts of the multiplication process:

{3,7-3 ,7,7+3 }

The following figures give some indication of the initial motion of the bi-variate probability distribution:

Table[Plot3D[Evaluate[P[x,y,(n/5) ]],{x,0,1},{y,0,1}, PlotRange->{0,12},Ticks->False],{n,0,25}]//TableForm

The motion appears to be "Lissajous-like" only in some very soft sense. But the situation gains sharpness when one looks to the  motion of the expected position of the quantum particle:

ExpPositionData=Table[NIntegrate[{x,y}P[x,y,(n/20)],{x,0,1},{y,0,1}],{n,0,100}];

QuantumOrbit=ListPlot[ExpPositionData,Joined->True, PlotRange->{{0,1}, {0,1}},PlotLabel->"Orbit traced by the expected position", AspectRatio->Automatic]

An excellent approxiation to such a figure—viewed simply as a mathematical object—could be produced by much simpler means:

ApproximatingSimpleLissajous=ParametricPlot[{.5+.175Sin[((2π)/7)t],.5+.175Sin[((2π)/3)t]}, {t,0,21},PlotRange->{{0,1}, {0,1}}, PlotStyle->Red]

Show[ApproximatingSimpleLissajous,QuantumOrbit]

X[t_] := x[t] /. {x0 -> 0, v ->1}
Y[t_] := x[t] /. {x0 -> 0, v -> (7/3)}
ParametricPlot[Evaluate[{X[t],Y[t]}], {t,0,20},GridLines->{{1},{1}}, Ticks->False, AxesLabel->{x,y}, LabelStyle->Large]

Of course, the trajectories traced by a classical particle confined to the interior of a square box are assembled from straight segments. In cases with

((Subscript[v, x])/(Subscript[v, y]))=(3/7)

We expect the moving quantum expectation value of position to trace an improved approximation to such a curve (with less rounded corners than were obtained previously) if we construct

Px[x,t]
Py[y,t]

by launching wavepackets that are intially well localized, and keep t so short that the localization remains well-preserved. The computational burden of such an approach is, however, great, so I will not attempt to illustrate the remark by explicit example. In such "hard" situations the moving function P[x, y, t] would itself—not just the associated expectation values, and except for complications at the reflection points—trace (for a while) a recognizable approximation to a Lissajous figure.

An alternative approach to the  quantum theory of Lissajous figures

Recently, Y. F. Chen, K, F, Huang & Y. P. Lan were led ("Localization of wave patterns on classical periodic orbits in a square billiard," Phys. Rev. E 66, 046215 (2002); "Quantum manifestations of classical periodic orbits in a square billiard: Formation of vortex lattices,"  Phys. Rev. E 66, 066210 (2002))—by appropriating the "Schwinger representation of the SU(2) algebra," a train of thought which I will not attempt to reproduce—to write (in my notation)

Clear[Ψ]

Ψ[x_,y_,m_,n_]:=2 Sin[m π x]Sin[n π y]

Ψ[x,y,k+1,n-k+1]

and from those functions to construct

Chen[x_,y_,n_, p_, q_,ϕ_]:=(1/(2^(n/2)))(∑_(k=0)^n)(Sqrt[Binomial[n,k]])(E^(I k ϕ))Ψ[p x,q y,k+1,n-k+1]

Chen[x,y,10, 7,3,0]

DensityPlot[((Chen[x,y,10, 7,3,0])^2),{x,0,1},{y,0,1}, PlotPoints->100]

Chen[x,y,26, 7,3,0]

DensityPlot[((Chen[x,y,26, 7,3,0])^2),{x,0,1},{y,0,1}, PlotPoints->100]

Vorticity

The preceding figure arises from deftly weighted superposition of a remarkably short list of eigenfunctions

Chen[x,y,26, 7,3,0]

Length[Expand[Chen[x,y,26, 7,3,0]]]

The point to which I would draw attention is that the associated probability density now does not have the form

Px[x]*Py[y]

The x and y variables are now correlated—not statistically independent.