Reading and digression on measurement

Pretty soon we will want to be using Dirac notation so I would suggest reading section 4.3 of Liboff, or something comparable, for that. Sections 4.4 (Hilbert spaces) and 4.5 (Hermitian objects) are also recommended.
Chapter 7 includes coverage of the harmonic oscillator. You could read that if you want. Our treatment in class should also be pretty complete. Griffiths uses similar notation. My thought was to use the Liboff notation for raising and and lowering “operators”, which can also be regarded as matrices. (In this context a matrix is a “representation” (or avatar) of an operator in a particular basis.) If the basis is well-chosen, the matrix representation can make the operator easier to understand and more intuitive.
Chapter 6 of Liboff covers time dependence. We’ll start on time dependence of quantum states in 1-D probably next week; our treatment should be self-contained, but you could read about it in advance of you like. Chapter 8 has a discussion of the finite square well, but I wouldn’t really recommend it. It seems overly complex and confusing. (Let me know if I am mistaken about that.)
Also of potential interest is section 5.1, and the subsection: Hilbert space interpretation has an interesting discussion, especially in the second to last paragraph which starts with “is there a chance…?” and includes the important phrase, "there is nothing in classical physics that is similar to this concept."
In this context, regarding measurement and its relationship to the state of a system, it's important to think about whether a measurement "leaves" the system in an eigenstate, or whether it actually destroys the system as we have conceived it, through interaction with a dissipative macroscopic system. Perhaps both are possible, depending on the nature of the measurement, but I think the latter is much more common.

Digression on measurement:

The question of what happens when a quantum particle, e.g., an electron, interacts with a macroscopic apparatus is rightfully thought to be important and significant. Though considerable effort and interest have been directed toward this issue, essentially it remains an unsolved problem. Recent papers, e.g., “Is measurement an emergent property,” P.W. Anderson PNAS, 2004?, speak to the extremely high level of difficulty and sophistication associated with this subject.

It is generally in the context of “measurement” that the concept of “wave-function collapse” arises. This may be something you have heard about somewhere? It is important to be clear that the time dependence invoked in casual discussions of wave-function collapse is distinct from that of the QM equations we study. The time dependence we will cover is based on the Schrodinger equation and comes from separation of variables. Wave-function collapse, as popularly conceived, is not, as far was we know, part of the Schrodinger equation-based formalism of QM, but rather is an addendum, an extra thing added on to address a perplexing conundrum . Ii could be described as conceptualizations or speculations regarding outcomes associated with a major class of unsolved problems.

The Schrodinger equation implies a particular sort of time dependence of all states arising from time evolving phase factors associated with each energy eigenstate (evolving at a rate proportional to its energy.) People have long been concerned that this Schrodinger wave-equation time dependence was not sufficient to describe what happens in a measurement. As I understand it, and i am no expert, wave-function collapse was hypothesized to address that concern. Although some very famous people have talked about it, it is not a tested or, as far as I know, testable theory.

I think that for a long time few people seriously considered the possibility that the time dependence of the Schrodinger equation could describe the sudden and seeming irreversible changes that occur during a measurement (involving a dissipative macroscopic object). Now, however, some people* have become less sure that time-dependence beyond that of the Schrodinger equation is needed to describe measurement. At any rate, the complexity of measurement systems, and perhaps other factors, tend to make these problems, for now, quite unsolvable.
* for example, N.D.Mermin