Lecture 11: Dispersion of the Gaussian and the Finite Well

The professor introduces the concept of a finite potential well, a theoretical model used in quantum mechanics to study bound states. Unlike real-world scenarios, this model assumes a potential that is piecewise constant, creating a simplified system to analyze. The lecture aims to solve for energy eigenfunctions and eigenvalues under specific boundary conditions, expecting discrete energies for bound states and continuous energies for scattering states.

The discussion delves into the technicalities of boundary conditions for wave functions in regions of constant potential. The professor explains how to solve the energy eigenvalue equation in these regions and emphasizes the importance of continuity in wave functions and their derivatives at boundaries. The behavior of wave functions in the presence of potential steps and singularities, such as delta functions, is also explored, highlighting the resulting discontinuities in derivatives.

The lecture continues with a detailed examination of the finite well potential, its parameters, and the quest to find bound states. The professor describes the mathematical approach to determine the wave function's behavior inside and outside the well, applying boundary conditions at infinity for normalizability and at the well's edges for continuity. The concept of bound states as energy eigenstates with exponential localization is introduced, contrasting with scattering states.

Building upon the previous discussions, the professor focuses on the derivation of the wave function for the finite well, considering both symmetric and antisymmetric solutions. The mathematical formulation involves solving differential equations with boundary conditions that ensure the wave function and its derivative are continuous. The lecture also addresses the simplification that arises from the potential's symmetry, leading to either even or odd wave functions.

The professor outlines the process of finding discrete energy eigenvalues for the finite well potential by solving a transcendental equation derived from the boundary conditions. The challenge of solving this equation numerically is introduced, with suggestions to use software like Mathematica. A graphical method for understanding the solutions is also presented, involving the use of dimensionless variables and the concept of a circle in a plot.

The lecture presents a graphical method to solve the transcendental equation for energy eigenvalues, using a circle representation to find intersections with the tangent function. The graphical technique reveals insights into the energy eigenstates and their dependence on the well's depth and width. The professor discusses how the number of bound states changes with the well's depth and width, emphasizing that there is always at least one bound state in a 1D finite well potential.

The professor explores the limiting cases of the finite well, such as the deep well limit that approximates an infinite well, and the behavior as the well becomes very shallow. The discussion includes the recovery of known results for the infinite square well and the implications of making the well wider or deeper on the energy eigenvalues. The analysis reinforces the understanding of bound states and their properties in the context of varying well parameters.

Towards the end of the lecture, the professor raises the question of the delta function potential, which is a singular limit of the finite well. The discussion explores the implications of such a potential on the wave function, suggesting the presence of a kink due to the discontinuity in the first derivative of the wave function. The lecture concludes with the anticipation of addressing the delta function potential in more detail in subsequent classes.

In conclusion, the professor summarizes the key findings of the lecture, including the graphical solution for energy eigenvalues and the properties of bound states in a finite well. The lecture ends with a preview of the next topic, which involves the delta function potential and its implications for quantum mechanical systems, such as the binding of atoms.