Question

Consider the three-dimensional problem of a mass m moving in a potential energy well given by V(r)=1/2kr².

a) Show using Schrödinger's equation that the spatial part of the quantum mechanical wavefunction separates into the product of three terms, each depending on only one of the three Cartesian coordinates and each satisfying the one-dimensional Schrödinger equation for a simple harmonic oscillator.
b) Find the 15 lowest possible energies of energy eigenstates of this system and the degeneracy of each of those energies.

Answer 1

The spatial part of the quantum mechanical wavefunction separates into three terms, each dependent on one of the Cartesian coordinates and satisfying the one-dimensional Schrödinger equation for a simple harmonic oscillator. The 15 lowest possible energies of energy eigenstates can be found using the one-dimensional Schrödinger equation for a simple harmonic oscillator potential, and the degeneracy of each energy level is 2n + 1.

Step-by-step explanation:

The spatial part of the quantum mechanical wavefunction separates into the product of three terms, each depending on only one of the three Cartesian coordinates (x, y, z), and each satisfying the one-dimensional Schrödinger equation for a simple harmonic oscillator. This separation is shown by substituting the wavefunction into Schrödinger's equation and recognizing that the resulting equation can be separated into three independent equations, each dependent on one coordinate.

As a result, the spatial part of the wavefunction can be represented as the product of three one-dimensional harmonic oscillator wavefunctions, one for each coordinate.

To find the 15 lowest possible energies of energy eigenstates and their degeneracies, we solve the one-dimensional Schrödinger equation for a simple harmonic oscillator potential. The energy eigenvalues are given by E_n = (n + 1/2)ħω, where n is the quantum number and ω is the angular frequency of the oscillator. The degeneracy of each energy level is 2n + 1.