By largely qualitative arguments. Exercise 37 shows that a...

By largely qualitative arguments. Exercise 37 shows that a finite well can hold an nth state only if its depth $U_{0}$ is at least $h^{2}(n-1)^{2} / 8 m L^{2}$. Show that this result also follows from equation $(5-23)$ and the accompanying Figure $5.14 .$

Key Concepts

Finite Potential Well

In quantum mechanics, a potential well is a region in space where the potential energy is lower than in the surrounding region. A finite potential well differs from an infinite one in that its walls are of finite height, which means particles can tunnel into classically forbidden regions, and the spectrum of bound states is discrete and limited by the depth and width of the well.

Bound State Energy Quantization

For a particle in a finite potential well, only certain energy levels (bound states) are allowed. These energy levels are quantized due to the requirement that the wavefunction must satisfy the Schrödinger equation and appropriate boundary conditions at the edges of the well. The number of bound states is determined by the well's depth and width, linking the physical parameters of the well to the quantized energy values.

Boundary Condition Matching

The quantization of energy in a finite well arises from enforcing the continuity of the wavefunction and its derivative at the boundaries of the well. This matching of boundary conditions leads to a set of equations—often transcendental in nature—that must be satisfied by the allowed energy levels, ensuring a physically acceptable solution of the Schrödinger equation.

Transcendental Equation and Graphical Analysis

The transcendental equation (such as equation (5-23)) results from applying the boundary conditions in the finite potential well problem. Since these equations typically cannot be solved analytically for exact energy levels, graphical methods, such as plotting the functions involved (as illustrated in Figure 5.14), are used to visually identify the intersections that correspond to allowed energy states.

Threshold Potential Condition

The threshold condition, which specifies that the well’s depth must be at least a certain value (in this case, proportional to h²(n–1)²/(8mL²)), is critical for the existence of an nth energy level. This condition emerges from analyzing the behavior of the transcendental equation and the corresponding graphical representation, indicating the minimum potential depth required to support a given number of bound states in the finite well.

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