1. The problem statement, all variables and given/known data An isotropic harmonic oscillator has the potential energy function U = 0.5 k (x²+y²+z²). (Isotropic means that the force constant is the same in all three coordinate directions.) (a) Show that for this potential, a solution to the three dimensional time-independent Schrodinger equation is given by ψ = ψ(x)ψ(y)ψ(z). In this expression, ψ(x) is a solution to the one-dimensional harmonic oscillator Schrödinger equation. The functions ψ(y) and ψ(z) are analogous one-dimensional wave functions for oscillations in the y- and z-directions. Find the energy associated with this ψ. (b) From your results in part (a) what are the ground-level and ﬁrst-excited-level energies of the three dimensional isotropic oscillator? (c) Show that there is only one state (one set of quantum numbers) for the ground level but three states for the ﬁrst excited level. 2. Relevant equations 3. The attempt at a solution (Sorry for my poor English.) I dont know even how to approach to this problem. I considered that a function of several variables can be written as a product of three functions of a single variable each one. I called these functions ψ(x), ψ(y), ψ(z). Then I wrote the 3d Schrod. Eq. substituting for ψ(x,y,z), and I divided both sides by that product: -(ħ²/2m) ((1 / ψ(x)) ∂² ψ(x) / ∂x² + ...) + U(x,y,z) = E I've read that the energy E cannot depend on the values of x, y, or z. Then I guess each term [ -(ħ²/2m) ((1 / ψ(x)) ∂² ψ(x) / ∂x² ] + [ -(ħ²/2m) ((1 / ψ(y)) ∂² ψ(y) / ∂y² + ...) ] + ... must be equal to a correspondent constant that doesnt depend on the direction. I called such constants Kx, Ky and Kz. So ψ(x)Kx + ψ(y)Ky + ψ(z)Kz + U(x,y,z) = E. But I'm not sure about my answer. In fact, I think it's wrong.