An electron, trapped in a finite potential energy well such as that of Fig. 39-7, is in its state of lowest energy. Are (a) its de-Broglie wavelength, (b) the magnitude of its momentum, and (c) its energy greater than, the same as, or less than they would be if the potential well were infinite, as in Fig. 39-2?

The given data can be listed below as,

• The depth of the well is, $U_0$.
• The width of the well is, L.

The term ‘de-Broglie wavelength’ of a charged particle (electron) depends on the charged particle's momentum value. The relationship between the de-Broglie wavelength and the particle’s momentum is an inverse one.

The given problem state about the distinction between a finite and infinite potential energy well.

The expression of the de-Broglie wavelength of the electron can be expressed as follows:

$$\begin{gathered} \lambda =\frac{h}{p} \\ =\frac{h}{\sqrt{2m\left(KE\right)}} \end{gathered}$$

Here, $\lambda$is the de-Broglie wavelength, h is the Plank’s constant, p is the electron’s momentum, m is the mass of the electron, and KE is the kinetic energy of the electron.

The motion of the trapped electron in the finite potential well is restricted x to the -direction. The de-Broglie wavelength of the infinite potential well is greater than the de-Broglie wavelength of the finite well because the kinetic energy of the electron in an infinite potential well is less than the kinetic energy of the electron in a finite well.

Thus, the de-Broglie wavelength is greater than the finite well.

The expression of momentum of the electron can be expressed as follows:

$p=\sqrt{2m\left(KE\right)}$

Here, p is the electron’s momentum, m is the mass of the electron, and KE is the kinetic energy of the electron.

The magnitude of the electron trapped in the infinite potential well has less momentum than that of the electron trapped in the finite potential well.

Thus, the electron's momentum magnitude is less than the finite potential well.

The expression of the energy of the electron can be expressed as follows:

$E_n=\left(\frac{h^2}{8m{L}^2}\right)n^2$

Here, $E_n$is the energy of the electron, L is the length of the well, and n is the quantum number.

The energy consisting of the electron trapped in the infinite potential well would have less energy than the electron trapped in the finite potential well.

Thus, the electron's energy is less than the electron's energy in the finite potential well.