5. Consider a cubic infinite potential well of length L. (Assume that the particle is confined to stay withing 0 and L in all the directions.) (a) What is the ground state energy and the ground state wavefunction? (b) What is the next energy level? Are there more than one wavefunction with the same energy? If there are then write down the quantum numbers for all the degenerate states.
6. Write down the energies of the next 5 levels and the quantum numbers for all degenerate energy eigenstates.
Schrödinger's equation inside the cube is:
With the boundary conditions:
Equation (1.1) is a separable equation.
We assume that the wave function is a product of three univariate independent functions
In this case, the partial differentials become full differentials:
We divide both sides by XYZ
We can rewrite the equation as:
Now the expression on the left side is a function of x only, while the expression on the right is a function of . The variables are completely independent and (1.6) must be true for any . The only way that this can be satisfied is if both sides of (1.6) equal the same ...
The solution shows how to solve the Schrodinger's equation for a particle in a 3D cube from scratch.