Chapter 10, Problem 10.79E

Interpretation Introduction

Interpretation:

The degeneracies of all levels for a cubical box from the lowest-energy wavefunction, described by the set of quantum numbers $\left(1,1,1\right)$ to the wavefunction described by the quantum number set $\left(4,4,4\right)$ are to be stated.

Concept introduction:

The Schrödinger equation is used to find the allowed energy levels for electronic transitions in the quantum mechanics. It is generally expressed as follows.

$\stackrel{⌢}{\text{H}}\Psi =\text{E}\Psi$

Where,

• $\stackrel{⌢}{\text{H}}$ is the Hamiltonian operator.

• $\Psi$ is the wavefunction.

• $\text{E}$ is the energy.

The energy obtained after applying the operator on wavefunction is known as the eigen value for the wavefunction.

Answer to Problem 10.79E

The degeneracies of all levels for a cubical box from the lowest-energy wavefunction, described by the set of quantum numbers $\left(1,1,1\right)$ to the wavefunction described by the quantum number set $\left(4,4,4\right)$ are given below.

$\left(1,1,2\right),\left(2,1,1\right),\left(1,2,1\right)$

$\left(2,1,2\right),\left(2,2,1\right),\left(1,2,2\right)$

$\left(3,1,1\right),\left(1,3,1\right),\left(1,1,3\right)$

$\left(3,2,1\right),\left(2,3,1\right),\left(3,1,2\right),\left(2,1,3\right),\left(1,2,3\right),\left(1,3,2\right)$

$\left(3,2,2\right),\left(2,3,2\right),\left(2,2,3\right)$

$\left(4,1,1\right),\left(1,4,1\right),\left(1,1,4\right)$

$\left(3,1,3\right),\left(3,3,1\right),\left(1,3,3\right)$

$\left(4,2,1\right),\left(4,1,2\right),\left(2,1,4\right),\left(2,4,1\right),\left(1,2,4\right),\left(1,4,2\right)$

$\left(3,3,2\right),\left(3,2,3\right),\left(2,3,3\right)$

$\left(4,2,2\right),\left(2,4,2\right),\left(2,2,4\right)$

$\left(4,3,1\right),\left(4,1,3\right),\left(3,1,4\right),\left(3,4,1\right),\left(1,3,4\right),\left(1,4,3\right)$

$\left(1,1,5\right),\left(5,1,1\right),\left(1,5,1\right),\left(3,3,3\right)$

$\left(4,3,2\right),\left(4,2,3\right),\left(3,2,4\right),\left(3,4,2\right),\left(2,3,4\right),\left(2,4,3\right)$

$\left(4,4,1\right),\left(4,1,4\right),\left(1,4,4\right),\left(5,2,2\right),\left(2,5,2\right),\left(2,2,5\right)$

$\left(4,3,3\right),\left(3,4,3\right),\left(3,3,4\right)$

$\left(4,2,4\right),\left(4,4,2\right),\left(2,4,4\right)$

$\left(4,4,3\right),\left(4,3,4\right),\left(3,4,4\right),\left(6,2,1\right),\left(6,1,2\right),\left(2,1,6\right),\left(2,6,1\right),\left(1,2,6\right),\left(1,6,2\right)$

Explanation of Solution

The formula to calculate energy for an electron confined to a three dimensional box is given below.

$E=\frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}+\frac{n_z^2}{c^2}\right)$ …(1)

Substitute the values in the equation (1) for the three-dimensional box having same dimensions and $n$ values to be $1,1,1$ as follows.

$\begin{array}{rll}{E}_{1,1,1}& =& \frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{1}{a^2}+\frac{1}{a^2}+\frac{1}{a^2}\right)\\ & =& \frac{3{\hslash }^2{\pi }^2}{2m{a}^2}\end{array}$

Similarly, other values for the energy by changing quantum numbers are calculated as follows.

$\begin{array}{rll}{E}_{1,2,1}& =& \frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{1}{a^2}+\frac{4}{a^2}+\frac{1}{a^2}\right)\\ & =& \frac{6{\hslash }^2{\pi }^2}{2m{a}^2}\end{array}$

$\begin{array}{rll}{E}_{2,1,1}& =& \frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{4}{a^2}+\frac{1}{a^2}+\frac{1}{a^2}\right)\\ & =& \frac{6{\hslash }^2{\pi }^2}{2m{a}^2}\end{array}$

$\begin{array}{rll}{E}_{1,1,2}& =& \frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{1}{a^2}+\frac{1}{a^2}+\frac{4}{a^2}\right)\\ & =& \frac{6{\hslash }^2{\pi }^2}{2m{a}^2}\end{array}$

The values of energies obtained for higher values of $n$ are as follows.

$\begin{array}{rll}{E}_{2,2,2}& =& \frac{{\hslash }^2{\pi }^2}{2m}\left(\frac{4}{a^2}+\frac{4}{a^2}+\frac{4}{a^2}\right)\\ & =& \frac{12{\hslash }^2{\pi }^2}{2m{a}^2}\end{array}$

$E_{2,1,2}=\frac{9{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{1,2,2}=\frac{9{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{2,2,1}=\frac{9{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,1,1}=\frac{11{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,1,2}=\frac{14{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,2,2}=\frac{17{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,3,2}=\frac{22{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,3,3}=\frac{27{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,1,1}=\frac{27{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,1,1}=\frac{18{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{1,5,1}=\frac{27{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{1,1,5}=\frac{27{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{3,3,1}=\frac{19{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,2,1}=\frac{21{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,2,2}=\frac{24{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,3,1}=\frac{26{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,4,1}=\frac{33{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,2,2}=\frac{33{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,3,3}=\frac{34{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,3,1}=\frac{35{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,4,2}=\frac{36{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,3,2}=\frac{38{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{6,1,1}=\frac{38{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,4,3}=\frac{41{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{6,2,1}=\frac{41{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,4,1}=\frac{42{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,3,3}=\frac{33{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{6,2,2}=\frac{44{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{5,4,2}=\frac{45{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{6,3,1}=\frac{46{\hslash }^2{\pi }^2}{2m{a}^2}$

$E_{4,4,4}=\frac{48{\hslash }^2{\pi }^2}{2m{a}^2}$

The set of quantum numbers degenerate wavefunctions are as follows:

$\left(1,1,2\right),\left(2,1,1\right),\left(1,2,1\right)$

$\left(2,1,2\right),\left(2,2,1\right),\left(1,2,2\right)$

$\left(3,1,1\right),\left(1,3,1\right),\left(1,1,3\right)$

$\left(3,2,1\right),\left(2,3,1\right),\left(3,1,2\right),\left(2,1,3\right),\left(1,2,3\right),\left(1,3,2\right)$

$\left(3,2,2\right),\left(2,3,2\right),\left(2,2,3\right)$

$\left(4,1,1\right),\left(1,4,1\right),\left(1,1,4\right)$

$\left(3,1,3\right),\left(3,3,1\right),\left(1,3,3\right)$

$\left(4,2,1\right),\left(4,1,2\right),\left(2,1,4\right),\left(2,4,1\right),\left(1,2,4\right),\left(1,4,2\right)$

$\left(3,3,2\right),\left(3,2,3\right),\left(2,3,3\right)$

$\left(4,2,2\right),\left(2,4,2\right),\left(2,2,4\right)$

$\left(4,3,1\right),\left(4,1,3\right),\left(3,1,4\right),\left(3,4,1\right),\left(1,3,4\right),\left(1,4,3\right)$

$\left(1,1,5\right),\left(5,1,1\right),\left(1,5,1\right),\left(3,3,3\right)$

$\left(4,3,2\right),\left(4,2,3\right),\left(3,2,4\right),\left(3,4,2\right),\left(2,3,4\right),\left(2,4,3\right)$

$\left(4,4,1\right),\left(4,1,4\right),\left(1,4,4\right),\left(5,2,2\right),\left(2,5,2\right),\left(2,2,5\right)$

$\left(4,3,3\right),\left(3,4,3\right),\left(3,3,4\right)$

$\left(4,2,4\right),\left(4,4,2\right),\left(2,4,4\right)$

$\left(4,4,3\right),\left(4,3,4\right),\left(3,4,4\right),\left(6,2,1\right),\left(6,1,2\right),\left(2,1,6\right),\left(2,6,1\right),\left(1,2,6\right),\left(1,6,2\right)$

Conclusion

The degeneracies of all levels for a cubical box from the lowest-energy wavefunction, described by the set of quantum numbers $\left(1,1,1\right)$ to the wavefunction described by the quantum number set $\left(4,4,4\right)$ are given below.

$\left(1,1,2\right),\left(2,1,1\right),\left(1,2,1\right)$

$\left(2,1,2\right),\left(2,2,1\right),\left(1,2,2\right)$

$\left(3,1,1\right),\left(1,3,1\right),\left(1,1,3\right)$

$\left(3,2,1\right),\left(2,3,1\right),\left(3,1,2\right),\left(2,1,3\right),\left(1,2,3\right),\left(1,3,2\right)$

$\left(3,2,2\right),\left(2,3,2\right),\left(2,2,3\right)$

$\left(4,1,1\right),\left(1,4,1\right),\left(1,1,4\right)$

$\left(3,1,3\right),\left(3,3,1\right),\left(1,3,3\right)$

$\left(4,2,1\right),\left(4,1,2\right),\left(2,1,4\right),\left(2,4,1\right),\left(1,2,4\right),\left(1,4,2\right)$

$\left(3,3,2\right),\left(3,2,3\right),\left(2,3,3\right)$

$\left(4,2,2\right),\left(2,4,2\right),\left(2,2,4\right)$

$\left(4,3,1\right),\left(4,1,3\right),\left(3,1,4\right),\left(3,4,1\right),\left(1,3,4\right),\left(1,4,3\right)$

$\left(1,1,5\right),\left(5,1,1\right),\left(1,5,1\right),\left(3,3,3\right)$

$\left(4,3,2\right),\left(4,2,3\right),\left(3,2,4\right),\left(3,4,2\right),\left(2,3,4\right),\left(2,4,3\right)$

$\left(4,4,1\right),\left(4,1,4\right),\left(1,4,4\right),\left(5,2,2\right),\left(2,5,2\right),\left(2,2,5\right)$

$\left(4,3,3\right),\left(3,4,3\right),\left(3,3,4\right)$

$\left(4,2,4\right),\left(4,4,2\right),\left(2,4,4\right)$

$\left(4,4,3\right),\left(4,3,4\right),\left(3,4,4\right),\left(6,2,1\right),\left(6,1,2\right),\left(2,1,6\right),\left(2,6,1\right),\left(1,2,6\right),\left(1,6,2\right)$