Canonical Conjugates and Fourier Transforms in Classical Physics

[Mark Spearman]

In quantum mechanics one can convert the wave function of one variable into the wave function of its conjugate pair (e.g., momentum and coordinate) using a Fourier transform.

Now consider the classical case. Suppose there is a particle in a potential well with insufficient energy to escape thereby resulting in a cyclic trajectory for the coordinate
q(t). What does the Fourier transform of q(t) over its period tell us about its momentum?

Perhaps it tells us nothing. Consider the simple case of a (classical) particle in a box with initial velocity of v and mass m. The periodic behavior of q(t) is a triangle wave while the momentum will be p = mv, reversing instantaneously at the walls of the well. The Fourier transform of a triangle wave is a squared sinc curve. It would appear that the distribution of p would be two delta functions, one at p and the other at -p.

So, does the Fourier transform tell us anything about conjugate pairs in classical physics?

Thanks,

Mark

 

[Studiot]

Good evening, Mark.

Can I commend to you the book about just this subject:

The Use of Integral Transforms

Ian Sneddon

He treats lots of transforms and applications as well as Fourier eg the Mehler-Fock transform in fracture mechanics.

go well