Dave Auckly
Coherent States, the propagator for the simple harmonic oscillator and
the Index Theorem

Title:
This term we have considered some background material in quantum
mechanics. Our motivation for this has been to derive a closed form of
the propagator for the simple harmonic oscillator. The result is quoted
in most textbooks on quantum mechanics and when it is derived, most
sources pull out a formula that is not commonly known. Our proof will
derive the formula using techniques that we have motivated without
referring to any formula from special functions. Mathematically, this is
a closed form solution for a certain heat equation on the real line.  In
some sense this is the universal heat equation one needs to understand
in order to compute the expected dimension of the space of solutions to
any elliptic system of PDEs. The seminal result in the area is called
the Atiyah Singer Index theorem. We are slowly presenting a proof and
statement of this theorem. The proof will pass from elliptic equations
to heat equations, then to wave equations before finally getting back to
elliptic equations.