Problem

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Imagine a box with zero potential enclosed in dimensions . Outside the box is the region where the particle’s wavefunction does not exist. Hence, the potential outside the box be infinite.

Boundary Conditions

Obtain the wavefunction of the particle in the box.

Solution

In quantum mechanics, wavefunctions are found by solving the Schrödinger equation. 

Use the separation of variables method to solve this partial differential equation (PDE). Consider the ansatz

with the partition of the total energy into its rightful components

.

Decompose the PDE into three ordinary differential equations (ODE):

where is the wavenumber, where is the mode.

Likewise,

The new boundary conditions for the decomposed ODEs is:

.

Solve the individual ODEs:

Apply the boundary conditions for to the solutions to arrive at the conclusion

.

Normalize the wavefunctions by performing the integral 

and apply the boundary condition .

Thus .

Repeat the normalization procedure for the and components.

Combine the solutions of all components of the ansatz to assemble the wavefunction

.
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