Question

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U₀ is increased) until essentially infinite.

Short answer

Wavelength will reduce & energy of the particle in the ground state will increase when the walls of the well are made higher.

Step-by-step solution

Step-1: Penetration Depth

The penetration depth is the reciprocal of a factor $\left(\alpha \right)$, which represents how far the wave function, representing the particle inside the well, extends in the classically forbidden region outside the well. It is given as-

$$\begin{gathered} \mathit{\delta }\mathbf{=}\frac{\mathbf{1}}{\mathbf{\alpha }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{\hslash }}{\sqrt{\mathbf{2}\mathbf{m}\left(U_o-E\right)}}\mathbf{}\mathbf{}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(1\right) \end{gathered}$$

Here, m and E are the mass and energy of the particle inside the well,role="math" localid="1660113324667" ${\mathbf{U}}_{\mathbf{o}}$is the height of the well, and$\mathit{\hslash }$is the modified Planck’s constant.

Step-2: Wave Function

For a particle in the ground state, equation (1) states that the penetration depth is inversely proportional to the square root of the difference in the height of the well and the energy of the particle. As the ground state energy of the particle remains constant, on increasing the walls of the well, wave function in the ground state will penetrate less and less into the classically forbidden region outside of the walls and when the well is infinitely high, the penetration depth is zero.

Step-3: Energies

On increasing the height of the well, the penetration depth decreases. This results in a decrease in the wavelength of the wave inside the well. Hence, the energy increases.

Thus, the penetration of wave function in the forbidden region decreases, and the energy of the wave increases, increasing the height of the well.