Particle in a Cube: Find Wavefunction w/Initial Conditions

[chill_factor]

Homework Statement

Problem 13.7.17 in Mathematical Methods in Physical Sciences:

Find the wavefunction of a particle in a cube, referring to 13.3.6.

0 < x < L, 0 < y < L, 0 < z < L

13.3.6:

Find the wavefunction of a particle in a square 0 < x < L, 0 < y < L. Assume V = 0.

Homework Equations

-(hbar)^2/2m * Laplacian(ψ) = i(hbar)*∂(ψ)/∂t

The Attempt at a Solution

Use separation of variables.

ψ = U(x,y,z)T(t)

Substitute UT into the equation and then divide both sides by UT to separate it into time dependent and time independent parts.

(hbar)^2/2m * Laplacian(U) - E*U = 0
i(hbar)*∂(ψ)/∂t = T

Solve the time dependent ordinary differential equation for T:

T = exp(-iEt/hbar)

If the time independent Schrödinger equation was in 1-D, it would be:

-(hbar)^2/2m * ∂(U)/∂x = E*U

Assume E = k^2, where k^2 = 2Em/(hbar)^2

∂(U)/∂x = -k^2*U

U must be a sin or cos function in terms of U(x) but due to boundary conditions that it must be 0 at x=0 and x=L, it cannot be cos which would be nonzero at x = 0.

U = sin(kx), k = n∏/L where n = 1,2,3...

By analog with the 1-D case, the 3-D solutions should be:

Ux = sin k1 * x
Uy = sin k2 * y
Uz = sin k3 * z

with the constants K being all a constant (n,m,p) times ∏/L .

The final solution is then ψ = UxUyUzT = Ʃ A(nmp)sin(k1x)sin(k2y)sin(k3z)exp(-iEt/hbar)

Now we attempt to use initial conditions to set up a triple Fourier series and find A(nmp) where the 1-D analog would be the Fourier series

A(n) = (2/L) * ∫(initial condition functions) sin(kx)dx from 0 to L.

The problem does NOT give initial conditions so I have no idea how to solve the problem now. What can I possibly assume for the initial conditions such that I can obtain a solution?

 

[diazona]

Given that no initial conditions are specified, it's probably asking for a generic solution, which you have.