Here we will solve the three dimensional particle in a box. To describe the system, we imagine a box with zero potential enclosed in dimensions , , . Outside the box is the region where the particle’s wavefunction does not exist. Hence, we let the potential outside the box be infinite (which means impossible to exist in physics).
As always, wavefunctions are found by solving the Schrödinger equation.
We use the separation of variables to solve this partial differential equation (PDE). Consider the ansatz:
We also partition the total energy into its rightful components:
Now we decompose the PDE into three ordinary differential equations (ODE):
where is the wavenumber, where is the mode.
For the and variables, we replace the above equation with and .
Solving the individual ODEs we find that
By the boundary conditions:
We normalize the wavefunctions by performing the integral
and applying the boundary condition .
Combine the solutions of all components to the ansatz yields
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