Show that the equation is separable

[leonne]

Homework Statement

This is a physics problem,
for which if the following three dimensional potentials would Schrodinger equation be separable

V=x²y + sin(z)
V= x² +y +tan^-1 (z^1/2)

Homework Equations

(-h²/2m)(d²ψ/dx² + d²ψ/dy² + d²ψ/dz² ) +v(x,y,z)ψ=Eψ

The Attempt at a Solution

Not really sure what to do. It says that the second one is and the first is not, but not sure what to do
in the book it goes through the steps when v=0 so would i use -h²/2m (1/X d²X)/dx²)=Ex and just add the v(x) to this and then the same for v(y)?

 

[mighty2000]

I would first clarify the question by asking for more information. What exactly do you mean by "separable" in this context? Are you asking if the Schrodinger equation can be solved analytically for these potentials, or if the potentials are separable in terms of variables (i.e. is it possible to write the wavefunction as a product of functions of x, y, and z)? Additionally, what are the boundaries of the potential and what are the energy levels being considered?

Once these questions are answered, I would approach the problem by plugging in the given potentials into the Schrodinger equation and seeing if it can be solved analytically. For the first potential, it may be possible to solve the equation for specific energy levels, but it is unlikely that it can be solved for all energy levels. For the second potential, it may be possible to separate the variables and solve the equation analytically for all energy levels.

If the question is asking about separability in terms of variables, I would try to write the wavefunction as a product of functions of x, y, and z and see if it satisfies the Schrodinger equation. If it does, then the potential is separable in terms of variables. If not, then it is not separable.

In any case, further clarification and information would be needed to fully answer this question.