Lecture 15: Eigenstates of the Angular Momentum Part 1

The professor begins by acknowledging the support of the audience, emphasizing the importance of donations for the continuation of MIT OpenCourseWare. The audience is informed about the upcoming midterm and the posting of a problem set with a revised due date. The professor suggests using the problem set as a review for the exam and highlights the format of the exam, which will consist of short answers and computations similar to the problem sets. A practice exam will be made available, and the professor recommends using it within a time constraint to gauge readiness. The connection between problem-solving practice and exam performance is stressed, and additional resources on OCW are suggested for further study. The professor also addresses a question about the exam's scope, confirming it will cover the entire semester's material.

The session continues with a review of quantum mechanics concepts, focusing on the implications of commutator relationships between energy and other operators. The professor explains how certain commutation relations indicate a ladder structure in the energy spectrum, allowing for the prediction of evenly spaced energy levels. The discussion then shifts to operators that commute with the energy operator, leading to the existence of simultaneous eigenfunctions and the possibility of multiple states sharing the same energy eigenvalue. The professor uses the example of a free particle to illustrate these concepts and contrasts it with the harmonic oscillator, which lacks such degeneracy due to its non-commutative nature with momentum. The lecture wraps up with a deeper dive into the meaning of eigenfunctions being complete and the relationship between complete bases and degeneracy in energy levels.

The lecture progresses to the exploration of quantum mechanics in higher dimensions, starting with a transition from a one-dimensional tricycle to a three-dimensional Yamaha, highlighting the persistence of fundamental dynamics. The conversation centers on the impact of dimensions on the principles discussed earlier, particularly the effect on the commutation relations and the potential changes in the understanding of quantum systems. The professor also introduces the concept of unitary operators and their relationship with energy operators, emphasizing the role of symmetries in the form of unitary transformations. Examples of translation and time evolution operators are given, and the connection between symmetries, conserved quantities, and the associated Hermitian operators is established.

The professor delves into the specifics of quantum mechanics in three dimensions, discussing the energy operator and its relation to position and momentum. The non-commutative nature of position and momentum operators is highlighted, leading to the uncertainty principle. The lecture explains that while position and momentum operators for different dimensions commute with each other, they do not share simultaneous eigenfunctions due to the uncertainty principle. The professor also introduces the concept of the Schrödinger equation in 3D, using vector notation and gradient operators to describe the wave function's time evolution. The importance of choosing appropriate coordinate systems for problem-solving is underscored, with a promise to explore the Laplacian in various coordinate systems.

The lecture focuses on the free particle in three dimensions, starting with the construction of energy eigenfunctions using the Schrödinger equation. The professor demonstrates the separability of the equation and constructs solutions in Cartesian coordinates. The solutions are expressed as products of one-dimensional wave functions, reflecting the particle's momentum in each direction. The concept of superposition is introduced to account for all possible momentum directions, leading to the understanding that not all solutions are of the simple product form assumed initially. The lecture concludes with a discussion about the normalizability of the wave functions and the implications of choosing oscillating solutions for the eigenfunctions.

The professor explores the concept of degeneracy in quantum systems, particularly in the context of a free particle in three dimensions. The discussion revolves around the infinite number of directions a particle can move in, each with the same energy, leading to a vast degeneracy of energy eigenfunctions. The lecture emphasizes the role of symmetry in causing degeneracies, as the system's invariance under rotation allows for states with the same energy but different momenta. The professor also contrasts this with the non-degenerate bound states in one dimension and sets the stage for examining the 3D harmonic oscillator to further investigate degeneracy in bound systems.

The lecture continues with an analysis of the 3D harmonic oscillator, focusing on the separability of the energy eigenvalue equation and its similarity to the 1D harmonic oscillator problem. The professor constructs the energy eigenfunctions by taking products of one-dimensional harmonic oscillator eigenfunctions for each spatial dimension. The energy levels are expressed as sums of the individual dimension's energy contributions. The discussion then turns to the degeneracy of these energy levels, highlighting the presence of degeneracies even in bound states in three dimensions. The professor illustrates this with examples of different excitation numbers and their corresponding energies, showing that states with different excitation patterns can share the same energy.

The professor investigates the pattern of degeneracies in the 3D harmonic oscillator, noting the increase in degeneracy with energy levels. The audience suggests a connection to number theory, prompting the professor to discuss the degeneracy numbers for various energy levels. A formula for the degeneracy is proposed, which is confirmed to hold for the given examples. The lecture emphasizes the deep connection between symmetries and degeneracies, explaining that degeneracies arise due to the system's rotational symmetry. The professor also points out that without symmetry, degeneracies would not exist, and that the presence of degeneracies can hint at underlying symmetries in the system.

The lecture concludes with an introduction to angular momentum, prompted by the discussion of rotational symmetry and its connection to degeneracy. The professor outlines the plan to study rotations and their generators, the angular momentum operators, in the following lectures. The audience's question about the symmetry explaining the growth of degeneracy is acknowledged, and the professor agrees to address it later. The importance of angular momentum and its addition in quantum mechanics is highlighted, with the professor sharing his realization of its significance in the field.

The professor begins by defining the angular momentum operator in quantum mechanics, drawing an analogy with its classical counterpart. The components of the angular momentum operator are derived, and concerns about operator ordering are addressed. The lecture continues with the expression of these operators in Cartesian and spherical coordinates, highlighting the simplicity of the Lz operator in spherical coordinates. The professor also introduces the concept of angular momentum squared and its significance in the Hamiltonian. The dimensions of angular momentum and the role of h bar as a dimensionless constant are discussed, leading to an exploration of the eigenfunctions of the angular momentum operators.

The lecture delves into the commutators of the angular momentum operators, revealing their simplicity despite initial complexity. The professor demonstrates the calculation of the commutator between Lx and Ly, leading to the discovery that the commutators cycle through the components of angular momentum. The implications of these commutators for the eigenfunctions of the angular momentum operators are discussed, highlighting the inability to find simultaneous eigenfunctions for Lx and Ly due to their non-commutative nature. However, the existence of simultaneous eigenfunctions for Lx (or Ly or Lz) and L squared is confirmed, setting the stage for further exploration of the angular momentum system.

The professor introduces the concept of a complete set of commuting observables, necessary to fully specify the state of a quantum system. The discussion revolves around the criteria for a set of operators to be considered complete, emphasizing that they must all commute with each other and that their eigenvalues must uniquely identify the system's state. The lecture applies this concept to the angular momentum system, identifying Lz and L squared as a complete set of commuting observables for angular momentum. The professor also addresses a question about the commutativity of L squared and Lx, attributing it to a brute force computation that students will perform in their problem set.

The lecture introduces ladder operators L plus and L minus, constructed from Lx and Ly, and explores their commutators with Lz and L squared. The professor