Question

Problems that require solving the three-dimensional schrödinger equation can often be reduced to related one-dimensional problems. an example of this would be the particle in a cubical box. consider a cubical box with rigid walls (i.e., u(x,y,z)=∞ outside of the cube) and edges of length l. the general solution for this problem is

Answer 1

The wavefunction for a particle in a one-dimension box is a well-known problem, which has as a solution:

Ψ(x) = √(2/l) sin (nπx/l)

When the box has three dimensions, the general solution is simply the multiplication of the solutions for each dimension, therefore:

Ψ(x) = K · sin (n_x·π·x / l) · sin (n_y·π·y / l) · sin (n_z·π·z / l)