Lecture 7: More on Energy Eigenstates

The script begins with an introduction to the topic of quantum mechanics, focusing on the concept of eigenfunctions and their significance in representing physical observables. It explains the dot product for functions, which is analogous to the dot product for vectors, and how it is used to calculate a number from two functions. The professor emphasizes the importance of eigenfunctions as they describe states with definite values of an observable, such as position, momentum, and energy. The script also discusses the orthogonality and normalization of these eigenfunctions, which are key properties in quantum mechanics.

This paragraph delves into the concept of degenerate eigenfunctions, where two different eigenfunctions can have the same eigenvalue. The professor explains that this is not only possible but also has physical implications. It introduces the idea that while eigenfunctions corresponding to different eigenvalues are orthogonal, degenerate eigenfunctions are not necessarily so. The spectral theorem is mentioned, which states that even with degeneracies, it's possible to find a set of orthogonal eigenfunctions, forming a basis for the space of functions.

The script discusses the ability to expand any function in terms of eigenfunctions of an observable, such as position or momentum. It uses the Fourier theorem to explain the expansion in terms of momentum eigenfunctions and touches on the physical interpretation of these expansions. The paragraph highlights the mathematical and physical significance of expressing any state as a superposition of states with definite values of an observable.

The focus shifts to energy eigenfunctions and their unique properties, particularly how they evolve over time. The professor explains that while energy eigenfunctions do not represent states with definite positions or momenta, they do have a simple time evolution described by the Schrödinger equation. The paragraph also addresses the discrete and continuous nature of energy eigenvalues in different physical systems and the implications for the expansion of wave functions.

This section discusses the time evolution of quantum states, especially those that are not energy eigenfunctions. The professor explains that while energy eigenfunctions evolve with a simple phase factor, a general state's evolution is more complex. The paragraph emphasizes the importance of understanding the time evolution of quantum states for solving problems in quantum mechanics.

The script provides an example of the infinite square well to illustrate the concept of energy eigenfunctions. It describes the potential energy function and how the energy eigenfunctions for this system are sine waves that satisfy the boundary conditions. The paragraph explains the quantization of energy levels and the corresponding wave functions for the ground state and the first excited state.

This paragraph explores the probability distributions associated with different energy eigenfunctions of the infinite square well. It contrasts the classical expectation of a uniform probability distribution with the quantum mechanical reality, where the probability distribution is not uniform and has nodes. The professor raises questions about the implications of these findings and their relation to the physical properties of materials like diamonds.

The script invites students to ponder the differences between classical and quantum mechanical probability distributions, especially at low energies where quantum effects dominate. It challenges students to think about why certain quantum phenomena occur, hinting at the role of superposition and interference in quantum mechanics.

The final paragraph of the script is a review of the concepts discussed and their application to quantum mechanics problems. The professor uses clicker questions to test students' understanding of eigenfunctions, time evolution, and the properties of quantum states. The questions cover topics such as the commutator of position and momentum operators, the inner product, and the behavior of wave functions.

The script concludes with a discussion on the qualitative behavior of energy eigenfunctions for an arbitrary potential. It explains how to derive the differential equation for energy eigenfunctions and how to interpret the second derivative in terms of curvature. The professor guides students through the process of estimating the qualitative features of wave functions based on the given potential.

This section provides a detailed analysis of how the curvature of a wave function, as indicated by the second derivative, can be used to predict the behavior of energy eigenfunctions. It explains that in classically allowed regions, the wave function should be sinusoidal, while in classically forbidden regions, it should be exponential. The professor uses this concept to sketch the expected wave function for a given energy and potential.

The script highlights the importance of boundary conditions in determining the physicality of a wave function. It demonstrates that a wave function that diverges to infinity is non-physical and must satisfy conditions such as vanishing at infinity for normalizability. The discussion also touches on the discrete nature of energy levels in systems with classically forbidden regions.

The final paragraph of the script wraps up the current discussion and hints at future topics. It leaves students with a teaser about the discrete nature of the hydrogen spectrum and the reasons behind it, setting the stage for the next class.