91影视

There is a 3D infinite cubic well.

The energy of a particle of mass m in a 3D infinite well of sides $L_x$, $L_y$and $L_z$ is

$E=\frac{{蟺}^2{\mathrm{鈩�}}^2}{2m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)$ ..... (I)

Here, $\mathrm{鈩�}$ is the reduced Planck's constant and $n_x$, $n_y$ and $n_z$ are the quantum numbers

(a)

For a cubic well, the energy in equation (I) reduces to

$E=\frac{{蟺}^2{\mathrm{鈩�}}^2}{2m{L}^2}\left(n_x^2+n_y^2+n_z^2\right)$ ..... (II)

Thus the energy when any one of $\left(n_x,n_y,n_z\right)$ is 5 and the rest two are 1 are the same. Hence the wave functions corresponding to $\left(n_x,n_y,n_z\right)=\left(5,1,1\right)$, $\left(n_x,n_y,n_z\right)=\left(1,5,1\right)$ and $\left(n_x,n_y,n_z\right)=\left(1,1,5\right)$ have the same energies. There are three wave functions having the same energy.

(b)

Let the length along X axis, that is$L_x$ be increased by $5\%$. The new length is then $\$1.05L\$$. The energy in equation (II) becomes

$E=\frac{{蟺}^2{\mathrm{鈩�}}^2}{2m{L}^2}\left(\frac{n_x^2}{{1.05}^2}+n_y^2+n_z^2\right)$

Thus the wave function corresponding to$\left(n_x,n_y,n_z\right)=\left(5,1,1\right)$will now have lower energy than the wave functions corresponding to $\left(n_x,n_y,n_z\right)=\left(1,5,1\right)$and$\left(n_x,n_y,n_z\right)=\left(1,1,5\right)$. The previous energy level will thus split into two separate energy levels.

(c)

The plot of the initial and final energy levels are as follows

(d)

There are still two wave functions corresponding to $\left(n_x,n_y,n_z\right)=\left(1,5,1\right)$ and $\left(n_x,n_y,n_z\right)=\left(1,1,5\right)$ that are in the same energy level. Thus a degeneracy of 2 is left. This can be removed if one of the lengths along x or y axis is slightly changed making the energy equation equal to equation (I).