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.
An overview of the spectral theorem for self-adjoint
operators. (Class handout.)
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.
of Quantum Mechanics.
The mathematical formalism of (non-relativistic) quantum mechanics.
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.
(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.
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.
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.