Outline and road map for 139a:

This post is intended to give you a idea of where we are going and help you prepare for each class and get a sense of the big picture and the details of what we are hoping to accomplish and cover in this class. I will continue to add to it as time goes on.

As mentioned previously, this class weaves together 3 themes which could be loosely characterized as:
1) quantum formalism, postulates, mathematical underpinnings and structure,
2) quantum physics in 1-dimension (mostly bound states),
3) quantum physics in more than 1-d (2 & 3), focusing particularly on the hydrogen atom with its interesting and unusual degeneracies, etc.

We will weave the formalism into 2) and 3) and deal with concrete examples for each new concept. Quantum physics provides the underpinning for most of physics, chemistry and more and, in terms of its impact both technology and fundamental understanding it is primarily the story of the electron. (Why?) That will be our approach. We endeavor to develop an integrated mathematical and intuitive understanding of this amazing, low-mass creature which, when confined, acquires significant kinetic energy. This simple and unusual trait plays a central role in many quantum phenomena of great importance.

We will start with the wave equation for electrons, which was discovered by Edwin Schrodinger in the 1920's and is now commonly know as the Schrodinger equation. We will never loose site of the essence of its being a wave equation. With that and other starting postulates we will do separation of variable, talk about time dependence, normalization and expectation values.

On Thursday, April 2, we will begin our investigations of 1-D QM. Specifically we will briefly study momentum eigenstates for free electrons, and then focus on the ground states of 4 different 1-D potentials and begin calculating expectation values of things like p, x, p^2, x^2, V and T (the kinetic energy).

The 4 potentials will be: infinite square well, harmonic oscillator, attractive delta-function and finite square well. (Note that two of them are actually limiting cases of the 4th one.) We will use the Sch.-wave eqn to solve for the g.s. in each case with appropriate boundary conditions. Interestingly, I think the boundary conditions appear quite differently in each case. You may want to look at the delta-function potential, and how the ground state emerges for that from the unusual boundary condition at the position of the delta-function. In general, it is the combination of boundary conditions, the wave equation and the normalization requirement that lead to the quantized energy levels from which quantum mechanics gets its name.

Then I think we should we look at momentum eigenstates and free-particle (gaussian) wave-packets, which are discussed, e.g., on page 160 of Liboff. This will help us understand confinement and differences between non-local and local (bound) states.

After that we can examine the excited bound states of our 1-D potentials. This will include exploring the use of raising and lowering operators for the harmonic oscillator. (Why are raising and lowering operators only used for the H.O.? Could they be developed and used for other postentials?)

With this concrete background we will be in a position to delve into the structure, formalism and meaning of quantum theory. Linear algebra, hermitian matrices, basis, linear expansion, superposition and inner-products will all be critical concepts here; we will explore their relevance to quantum theory. At this point we can also talk a bit about measurement and "non-commuting observables". x and p are examples of non-commuting observables. More examples will arise in a natural wasy when we begin to study 2 and 3-dimensional quantum systems, which have degenerate energy eigenstates arising from their symmetries, something that is largely absent in 1-d.
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