Lecture 17: More on Central Potentials

The script begins with an introduction to MIT OpenCourseWare, highlighting its mission to provide free, high-quality educational resources under a Creative Commons License. The professor then invites questions about angular momentum from previous lectures, leading to a detailed discussion on the peculiarities of half-integer spin states. These states, which cannot be represented by a wave function on a sphere, are previewed to be discussed in detail later in the course. The professor uses the analogy of a tiny bar magnet in a magnetic field gradient to hint at the intrinsic angular momentum of particles like electrons, setting the stage for a deeper exploration of quantum mechanics.

The professor delves into the relationship between angular momentum and magnetic moment, using the example of a charged sphere to explain how rotation generates a magnetic field. This concept is applied to the Stern Gerlach experiment, which measures the angular momentum of electrons by observing their deflection in a magnetic field gradient. The experiment reveals that electrons exhibit a quantized angular momentum, taking only specific values, and challenges classical notions of rotation. The discussion introduces the concept of 'spin' as an intrinsic form of angular momentum for particles like electrons, which is distinct from classical rotational angular momentum.

This section explores the historical context of the discovery of quantum angular momentum. The professor clarifies that the mathematical prediction of half-integer states preceded their physical understanding and experimental confirmation. The audience is reminded that the commutation relations leading to these states were observed mathematically before their physical significance was realized. The discussion also touches on the experimental observation of angular momentum being confined to specific values, leading to the concept of spin.

The script transitions into a discussion of three-dimensional quantum systems, with a focus on spherical coordinates. The professor introduces the Laplacian in spherical coordinates and highlights its significance in solving central potentials. The Laplacian is broken down into radial and angular components, with the angular component related to the square of the angular momentum operator. The section sets the stage for solving problems in 3D, including the exploration of spherically symmetric systems like the hydrogen atom.

The professor discusses the method of separation of variables to simplify the energy eigenvalue equation for a central potential. By recognizing that the energy operator and the angular momentum squared commute, the professor shows that it's possible to find common eigenfunctions for both. This leads to a simplification where the energy eigenvalue equation becomes a one-dimensional problem for the radial part of the wave function, with the angular part contributing a constant to the effective potential.

The script continues with the exploration of the effective potential in a central potential problem. The professor introduces a rescaling of the wave function to eliminate the radial dependence in the equation, leading to a cleaner one-dimensional energy eigenvalue equation. The effective potential is shown to consist of the original potential plus a term related to the angular momentum, which becomes significant near the origin and falls off rapidly with distance.

The professor discusses the concept of the angular momentum barrier, which arises from the kinetic energy associated with angular momentum. As particles approach the origin, their angular velocity and thus kinetic energy increase, creating a potential barrier. However, this barrier does not prevent decay in quantum systems like hydrogen, which remains stable due to other quantum mechanical effects.

The script addresses the boundary conditions for wave functions in central potentials, emphasizing that the wave function must vanish at the origin to avoid unphysical energy divergences. The professor also explains the degeneracy of energy levels with respect to the magnetic quantum number m, which is a consequence of rotational symmetry. The energy levels depend on the total angular momentum quantum number l but not on m, leading to 2l+1-fold degeneracy for each energy level.

The professor presents a simplified model of a spherical well potential to illustrate the application of the concepts discussed. By considering the special case of zero angular momentum, the professor derives the wave function and energy levels for the system. The solutions are found to satisfy boundary conditions at the origin and at the well's edge, leading to quantized energy levels. The wave functions are expressed in terms of spherical harmonics and a radial function, which is a sine function of the product of the angular frequency and the radial distance.

The script concludes with an introduction to the toy model for the hydrogen atom, which is an idealized version of the Coulomb potential. The professor emphasizes that this model, while not exact, provides a good first approximation to the behavior of hydrogen. The potential is derived from the assumption of point particles with specific charges and masses, leading to a simple inverse-square law. The professor also introduces the characteristic length and energy scales for the system, setting the stage for a more detailed analysis in subsequent lectures.

The professor applies dimensional analysis to the Coulomb potential problem, introducing a characteristic length scale and energy for the system. The script then outlines the method for solving the resulting differential equation, which involves asymptotic analysis and a series expansion. The solutions are expressed in terms of dimensionless variables, and the energy levels are related to integer quantum numbers. The professor also discusses the implications of the solutions for the observed spectrum of hydrogen, highlighting the agreement with the Rydberg formula and the unexpected degeneracy of energy levels.