Functional Analysis and Quantum Mechanics: Notes

Here are the handouts from the seminar, and some additional notes. I would appreciate any feedback on these that you read, especially on "A Pseudo-History of Quantum Mechanics". It is still incomplete, but is already longer than I would prefer, and I would really like to clarify (and shorten, if possible) some of the presentation.
  1. The Spectral Theorem.
    An overview of the spectral theorem for self-adjoint operators. (Class handout.)
  2. The Measurable Functional Calculus.
    An alternative method of extending the functional calculus from continuous tomeasurable functions, in case you don't like the appeal to the Riesz Representation Theorem.
  3. Summary of Quantum Mechanics.
    The mathematical formalism of (non-relativistic) quantum mechanics. (Class handout.)
  4. Quantum Dynamics: Examples.
    The dynamics of the quantum harmonic oscillator, and of a free particle, worked out according to the abstract formalism in the previous set of notes. (Class handout.)
  5. A (Very) Brief History of Quantum Mechanics.
    Why would anyone think that a physical observable should be represented mathematically by an operator? In my opinion the best way to appreciate the answer to this question is to see what Heisenberg did in 1926, reasoning by analogy from the equations of classical mechanics. This set of notes explains Heisenberg's argument.
  6. A Pseudo-history of Quantum Mechanics.
    The quantum theory developed between 1911 and 1926 is now known as the "old quantum theory". It was a somewhat ad-hoc mixture of classical and quantm arguments, and is hard to understand without a strong background in classical mechanics and electromagnetism. The "new quantum mechanics" that began with Heisenberg's paper is nearly self-contained, and potentially much easier to uderstand. This set of notes, still incomplete, is an attempt to motivate and explain the modern theory of quantum mechanics while bypassing the historical path through the old theory. It can be seen as an enlarged version of the "Brief History" notes.
  7. The HVZ Theorem.
    The goal of this set of notes is to give a self-contained proof of the HVZ Theorem (due to Hunziker, Van Winter and Zhislin), which determines the so-called "essential spectrum" of an N-particle hamiltonian. In order to do so, however, the notes first develop much of the basic functional-analytic theory of atomic and molecular hamiltonians, including Kato's fundamental theorem that such operators are essentially self-adjoint.

Kevin Mcleod
03/02/2011