Chapter 10, Problem 10.81E

Interpretation Introduction

Interpretation:

The degeneracies for all energies of a $\text{2-D}$ particle-in-a-box up to $n_x=n_y=8$ are to be stated. The plot of the energy levels in a graph is to be drawn.

Concept introduction:

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

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

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 10.81E

The degeneracies for all energies of a $\text{2-D}$ particle-in-a-box up to $n_x=n_y=8$ have been calculated and the plot of energy versus energy levels in a graph is shown below.

Explanation of Solution

The formula to calculate energy for $\text{2-D}$ box can be written as given below.

$E=\frac{h^2}{8m}\left(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}\right)$

Assuming $a=b$, the energy corresponding to lowest possible quantum numbers $\left(1,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,1}& =& \frac{h^2}{8m}\left(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}\right)\\ & =& \frac{h^2}{8m}\left(\frac{1^2}{a^2}+\frac{1^2}{a^2}\right)\\ & =& \frac{h^2}{8m}\left(\frac{1}{a^2}+\frac{1}{a^2}\right)\\ & =& \frac{2h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,2\right)\left(2,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,2}& =& \frac{5h^2}{8m{a}^2}\\ E_{2,1}& =& \frac{5h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,3\right)\left(3,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,3}& =& \frac{10h^2}{8m{a}^2}\\ E_{3,1}& =& \frac{10h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,4\right)\left(4,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,4}& =& \frac{17h^2}{8m{a}^2}\\ E_{4,1}& =& \frac{17h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,3\right)\left(3,2\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{2,3}& =& \frac{13h^2}{8m{a}^2}\\ E_{3,2}& =& \frac{13h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,4\right)\left(4,2\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{2,4}& =& \frac{20h^2}{8m{a}^2}\\ E_{4,2}& =& \frac{20h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(3,4\right)\left(4,3\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{3,4}& =& \frac{25h^2}{8m{a}^2}\\ E_{4,3}& =& \frac{25h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,5\right)\left(5,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,5}& =& \frac{26h^2}{8m{a}^2}\\ E_{5,1}& =& \frac{26h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,5\right)\left(5,2\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{2,5}& =& \frac{29h^2}{8m{a}^2}\\ E_{5,2}& =& \frac{29h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(3,5\right)\left(5,3\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{3,5}& =& \frac{34h^2}{8m{a}^2}\\ E_{5,3}& =& \frac{34h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,6\right)\left(6,1\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,6}& =& \frac{37h^2}{8m{a}^2}\\ E_{6,1}& =& \frac{37h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,6\right)\left(6,2\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{2,6}& =& \frac{40h^2}{8m{a}^2}\\ E_{6,2}& =& \frac{40h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(4,5\right)\left(5,4\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{4,5}& =& \frac{41h^2}{8m{a}^2}\\ E_{5,4}& =& \frac{41h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(3,6\right)\left(6,3\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{3,6}& =& \frac{45h^2}{8m{a}^2}\\ E_{6,3}& =& \frac{45h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,7\right)\left(7,1\right)\left(5,5\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,7}& =& \frac{50h^2}{8m{a}^2}\\ E_{7,1}& =& \frac{50h^2}{8m{a}^2}\\ E_{5,5}& =& \frac{50h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(4,6\right)\left(6,4\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{4,6}& =& \frac{52h^2}{8m{a}^2}\\ E_{6,4}& =& \frac{52h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,7\right)\left(7,2\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{2,7}& =& \frac{53h^2}{8m{a}^2}\\ E_{7,2}& =& \frac{53h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(3,7\right)\left(7,3\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{3,7}& =& \frac{58h^2}{8m{a}^2}\\ E_{7,3}& =& \frac{58h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(6,5\right)\left(5,6\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{6,5}& =& \frac{61h^2}{8m{a}^2}\\ E_{5,6}& =& \frac{61h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(1,8\right)\left(8,1\right)\left(4,7\right)\left(7,4\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{1,8}& =& \frac{65h^2}{8m{a}^2}\\ E_{8,1}& =& \frac{65h^2}{8m{a}^2}\\ E_{4,7}& =& \frac{65h^2}{8m{a}^2}\\ E_{7,4}& =& \frac{65h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(2,8\right)\left(8,2\right)$are calculated as shown below.

$\begin{array}{rll}{E}_{2,8}& =& \frac{68h^2}{8m{a}^2}\\ E_{8,2}& =& \frac{68h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(3,8\right)\left(8,3\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{3,8}& =& \frac{73h^2}{8m{a}^2}\\ E_{8,3}& =& \frac{73h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(5,7\right)\left(7,5\right)$are calculated as shown below.

$\begin{array}{rll}{E}_{5,7}& =& \frac{74h^2}{8m{a}^2}\\ E_{7,5}& =& \frac{74h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(4,8\right)\left(8,4\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{4,8}& =& \frac{80h^2}{8m{a}^2}\\ E_{8,4}& =& \frac{80h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(6,7\right)\left(7,6\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{6,7}& =& \frac{85h^2}{8m{a}^2}\\ E_{7,6}& =& \frac{85h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(5,8\right)\left(8,5\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{5,8}& =& \frac{89h^2}{8m{a}^2}\\ E_{8,5}& =& \frac{89h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(6,8\right)\left(8,6\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{6,8}& =& \frac{100h^2}{8m{a}^2}\\ E_{8,6}& =& \frac{100h^2}{8m{a}^2}\end{array}$

Similarly, energies corresponding to energy levels $\left(7,8\right)\left(8,7\right)$ are calculated as shown below.

$\begin{array}{rll}{E}_{7,8}& =& \frac{113h^2}{8m{a}^2}\\ E_{8,7}& =& \frac{113h^2}{8m{a}^2}\end{array}$

Similarly, energy corresponding to energy levels $\left(2,2\right)$ is calculated as shown below.

$E_{2,2}=\frac{8h^2}{8m{a}^2}$

Similarly, energy corresponding to energy levels $\left(3,3\right)$ is calculated as shown below.

$E_{3,3}=\frac{18h^2}{8m{a}^2}$

Similarly, energy corresponding to energy levels $\left(4,4\right)$ is calculated as shown below.

$E_{4,4}=\frac{32h^2}{8m{a}^2}$

Similarly, energy corresponding to energy levels $\left(6,6\right)$ is calculated as shown below.

$E_{6,6}=\frac{72h^2}{8m{a}^2}$

Similarly, energy corresponding to energy levels $\left(7,7\right)$ is calculated as shown below.

$E_{7,7}=\frac{98h^2}{8m{a}^2}$

Similarly, energy corresponding to energy levels $\left(8,8\right)$ is calculated as shown below.

$E_{8,8}=\frac{128h^2}{8m{a}^2}$

The plot of energy versus energy levels is shown in the figure 1.

Figure 1

Conclusion

The degeneracies for all energies of a $\text{2-D}$ particle-in-a-box up to $n_x=n_y=8$ have been calculated and the plot of energy versus energy levels is shown in figure 1.