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

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.

Wednesday, February 26, 2020 at 2:30pm

- Department
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