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The potential energy is the stored energy that depends upon the relative position of various parts of a system.

The given potential in the finite well is:

$\mathrm{U}=\left\{\begin{array}{l}\begin{array}{ll}\frac{{\mathrm{鈩�}}^2{\mathrm{k}}^2}{2\mathrm{m}}{\sec }^2\frac{\mathrm{kL}}{2}& (\mathrm{n}鈭�1)\mathrm{蟺}<\mathrm{kL}<\mathrm{苍蟺}\text{鈥�}\mathrm{n}=1,3,5,鈥�\end{array}\\ \begin{array}{ll}\frac{{\mathrm{鈩�}}^2{\mathrm{k}}^2}{2\mathrm{m}}{\csc }^2\frac{\mathrm{kL}}{2}& (\mathrm{n}鈭�1)\mathrm{蟺}<\mathrm{kL}<\mathrm{苍蟺}\text{鈥嬧赌嬧赌嬧赌嬧赌夆赌夆赌夆赌�}\mathrm{n}=2,4,6....\end{array}\end{array}\right.$

The formula for the potential for even $\mathrm{n}$is given by:

$\mathrm{U}=\frac{{\mathrm{鈩�}}^2{\mathrm{k}}^2}{2\mathrm{m}}{\csc }^2\frac{\mathrm{kL}}{2}\text{鈥�}(\mathrm{n}鈭�1)\mathrm{蟺}<\mathrm{kL}<\mathrm{苍蟺}\text{鈥�}鈥�\mathrm{...}\text{(1)}$

Here,$U$ is the potential, $\mathrm{鈩�}$is the reduced Planck's constant, $k$is expressed as constant,$m$ is the mass and $L$is the depth of the potential well.

The expression for a minimum value of $k$.

$\mathrm{k}=\frac{(\mathrm{n}鈭�1)\mathrm{蟺}}{\mathrm{L}}$

Here, $n$is the number of states.

Calculation:

Substitute$\frac{(\mathrm{n}鈭�1)\mathrm{蟺}}{\mathrm{l}}$for$k$in equation (1).

$\mathrm{U}=\frac{{\mathrm{鈩�}}^2}{2\mathrm{m}}\left(\frac{{(\mathrm{n}鈭�1)}^2{\mathrm{蟺}}^2}{{\mathrm{L}}^2}\right){\csc }^2\frac{\mathrm{L}}{2}\left(\frac{(\mathrm{n}鈭�1)\mathrm{蟺}}{\mathrm{L}}\right)$

$\mathrm{U}==\frac{{\mathrm{鈩�}}^2{\mathrm{蟺}}^2{(\mathrm{n}鈭�1)}^2}{2{\mathrm{mL}}^2}{\csc }^2\left(\frac{(\mathrm{n}鈭�1)\mathrm{蟺}}{2}\right)$

Substitute$1$ for the minimum value of${\csc }^2\left(\frac{(n鈭�1)蟺}{2}\right)$ in above equation.

$\mathrm{U}=\frac{{\mathrm{鈩�}}^2{\mathrm{蟺}}^2{(\mathrm{n}鈭�1)}^2}{2{\mathrm{mL}}^2}$

Substitute $\frac{\mathrm{h}}{2\mathrm{蟺}}$for$\mathrm{鈩�}$ in the above equation.

$\mathrm{U}=\frac{{\mathrm{h}}^2{\mathrm{蟺}}^2{(\mathrm{n}鈭�1)}^2}{8{\mathrm{mL}}^2}$

Thus, the potential energy of infinite well is$\frac{{\mathrm{h}}^2{\mathrm{蟺}}^2{(\mathrm{n}鈭�1)}^2}{8{\mathrm{mL}}^2}$ .