Figure 39-25 shows three infinite potential wells, each on an x axis. Without written calculation, determine the wave function $\mathit{\psi }$for a ground-state electron trapped in each well.

The given data can be listed below as,

• The length of the first well is, $L_1=2L$.
• The length of the second well is,$L_2=\frac{L}{2}$ .
• The length of the third well is, $L_3=L$.

The term ‘wave function’ for an electron that trapped in an infinite, one-dimensional specific potential well represents a relation between the length of the well and the quantum number.

The expression to calculate the wave function$\psi$ for a ground-state electron trapped in well (a) can be expressed as,

${\psi }_1=Asin\left(\frac{n\mathrm{\pi x}}{L_1}\right)$

Here,A is the normalization constant and${\psi }_1$ is the wave function for a ground-state electron trapped in well (a).

Substitute all the known values in the above equation.

${\psi }_1=Asin\left(\frac{n\mathrm{\pi x}}{2L}\right)$

Thus, the wave function for a ground-state electron trapped in well (a) is ${\psi }_1=Asin\left(\frac{n\mathrm{\pi x}}{2L}\right)$.

The expression to calculate the wave function$\psi$ for a ground-state electron trapped in well (b) can be expressed as,

${\psi }_2=A\sin \left(\frac{n\mathrm{\pi x}}{L_2}\right)$

Here,${\psi }_2$ is the wave function for a ground-state electron trapped in well (b).

Substitute all the known values in the above equation.

$$\begin{gathered} {\psi }_1=A\sin \left(\frac{n\mathrm{\pi x}}{{\displaystyle \frac{L}{2}}}\right) \\ =A\sin \left(\frac{2n\mathrm{\pi x}}{L}\right) \end{gathered}$$

Thus, the wave function for a ground-state electron trapped in well (b) is ${\psi }_2=A\sin \left(\frac{2n\mathrm{\pi x}}{L}\right)$.

The expression to calculate the wave functionfor a ground-state electron trapped in well (c) can be expressed as,

${\psi }_3=A\sin \left(\frac{n\mathrm{\pi x}}{L_3}\right)$

Here,${\psi }_3$ is the wave function for a ground-state electron trapped in well (c).

Substitute all the known values in the above equation.

$$\begin{gathered} {\psi }_2=A\sin \left(\frac{n\mathrm{\pi x}}{L}\right) \\ =A\sin \left(\frac{n\mathrm{\pi x}}{L}\right) \end{gathered}$$

Thus, the wave function for a ground-state electron trapped in well (c) is ${\psi }_3=A\sin \left(\frac{n\mathrm{\pi x}}{L}\right)$.