Extension to Infinite Well Three Dimensions (3D Box)

EXTENSION TO INFINITE WELL THREE DIMENSIONS (3D Box)

The solution of one-dimensional potential well is extended for a three-dimensional (3D) potential box.

In a three-dimensional potential box, the particle (electron) can move in any direction in space. Therefore, instead of one quantum number n, we have to use three quantum numbers, nx , ny and nz, corresponding to the three coordinate axes namely x, y and z respectively.

If a,b,c are the lengths of the box as shown in fig. 6.12 along x, y and z axes, then

If a = b = c as for a cubical box, then

The corresponding normalised wave function of the particle in the three dimension well is written as

From equations (1) and (2), we understand that several combinations of the three quantum numbers (nx, ny and nz) lead to different energy eigen values and eigen functions. 

Note: Three dimensional infinite potential well is an example of quantion dot. 

Example 

Suppose a state has quantum numbers

nx = 1, ny =1, nz = 2 

Then, nx2 + ny2 + nz2 = 12 + 12 + 22 = 1 + 1 + 4 = 6

Similarly, for a combination nx = 1, ny =2, nz = 1 and for a combination nx = 2, ny =1, nz = 1 

The corresponding wave functions is written as

Degeneracy

It is noted from equations (3) and (4) that, for several combinations of quantum numbers, we have the same energy eigen value but different eigen functions. Such a state of energy levels is called degenerate state.

The three combinations of quantum numbers, (112), (121) and (211), which give the same eigen value but different eigen functions are called 3-fold degenerate state.

Non-degenerate state: 

When only one wave function corresponds to the energy eigen value, such a state is called non-degenerate state.

Suppose