Solving Schrodinger's Equation for a Particle in an Infinite Box

[Boltzman Oscillation]

Homework Statement

Consider a particle of mass m, what is the total energy?

Relevant Equations

V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a
V(x,y,z) = 0 elsewhere

Firstly, since there is no condition for the z axis in the definition of the potential can I assume that V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a AND -inf<z<inf?

If so then drawing the potential I can see that the particle is trapped within a box with infinite height (if z is the vertical axis). Now I know that i can separate the Schrödinger equation into three parts, one with the x coordinates, one with the y coordinates, and one with the z coordinates. They are related by:

$$E_{total} = E_x + E_y + E_z$$
and
$$\psi_{total} = \psi_x * \psi_y * \psi_z$$
But when I try to solve the Schrödinger in the x coordinate case, i.e:
$$\frac{-h^2}{2m}\frac{\partial^2 \psi_x(x)}{\partial x^2} + V(x)\psi_x(x) = E_x\psi_x$$
would V(x) = 0 since V(x,y,z) = .5mw^2z^2? Can I not separate these potentials to begin with into three different functions? Am I doing this question right? Any help is appreciated.

 

[PeroK]

Homework Statement: Consider a particle of mass m, what is the total energy?
Homework Equations: V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a
V(x,y,z) = 0 elsewhere

Firstly, since there is no condition for the z axis in the definition of the potential can I assume that V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a AND -inf<z<inf?

If so then drawing the potential I can see that the particle is trapped within a box with infinite height (if z is the vertical axis). Now I know that i can separate the Boltzmann equation into three parts, one with the x coordinates, one with the y coordinates, and one with the z coordinates. They are related by:

$$E_{total} = E_x + E_y + E_z$$
and
$$\psi_{total} = \psi_x * \psi_y * \psi_z$$
But when I try to solve the Boltzmann in the x coordinate case, i.e:
$$\frac{-h^2}{2m}\frac{\partial^2 \psi_x(x)}{\partial x^2} + V(x)\psi_x(x) = E_x\psi_x$$
would V(x) = 0 since V(x,y,z) = .5mw^2z^2? Can I not separate these potentials to begin with into three different functions? Am I doing this question right? Any help is appreciated.

$V$ is a function of $z$ only, so it is already a separated function. Just separate out the potential into the z-equation.