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The Particle in a Cube

  1. Feb 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Problem 13.7.17 in Mathematical Methods in Physical Sciences:

    Find the wavefunction of a particle in a cube, refering 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.

    2. Relevant equations

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

    3. 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 schrodinger 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?
     
  2. jcsd
  3. Feb 12, 2012 #2

    diazona

    User Avatar
    Homework Helper

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