Chapter 41, Problem 41.44P

(a)

To determine

To show: For the potential $U\left(x,y,z\right)=\frac{1}{2}k\text{'}\left(x^2+y^2+z^2\right)$, $\psi ={\psi }_{n_y}\left(x\right){\psi }_{n_y}\left(y\right){\psi }_{n_z}\left(z\right)$ is a solution of three-dimensional time-independent Schrodinger equation, where ${\psi }_{n_x}$ is a solution to the one-dimensional harmonic oscillator Schrodinger equation with energy $E_{n_x}=\left(n_x+\frac{1}{2}\right)\sout{h}\omega$, ${\psi }_{n_y}$ and ${\psi }_{n_z}$ are the analogous one-dimensional wave functions for oscillations in the $y$-and $z$-directions and to determine the energy associated with $\psi$.

(b)

To determine

The ground level and first-excited level energies of the three-dimensional isotropic oscillator.

(c)

To determine

To show: There is only one state for the ground level but three states for the first excited level.