Monday Math 128

We have previously discussed the Fourier transform (here and here, especially). In this post, we noted that (using the symmetric angular convention) the space transform for an n dimensional space is

and the inverse is
.
We can also do the same for a vector field:

and
.
We note from vector calculus then that for a vector field, the components of the transform are the transforms of the components:
.

We also used integration by parts here to show that for a one-dimensional function f(x), with , the derivative has Fourier transform:
.
Similarly, we can use vector integration by parts for our multi-dimensional transforms. Working in three dimensions from here:
First, one form of the divergence theorem states:
,
where S is the boundary of the volume V, with outward normal.
Letting f=φψ, and using the gradient product rule ,

.
Letting , we see
.
and since , we have
.
Now, if as , φ(x) goes to zero faster than , then, as we expand the volume V to cover all space, the surface integral will go to zero, and we obtain
,
which means
,
in analogy to our one-dimensional rule.

Physics Friday 44

Quantum Mechanics and Momentum
Part 1: The De Broglie Relations

Planck’s relation tells us that the energy of a photon is proportional to the frequency: E=hν, where h is Planck’s constant. In terms of angular frequency ω, we use ω=2πν to find that , where is called the reduced Planck constant, or the Dirac constant.
Classical electrodynamics of an electromagnetic wave and the relativistic energy-momentum relation for a particle with zero rest mass both give the same result for the momentum of a photon: that , and so the momentum is
, where λ is the wavelength. Solving for wavelength, we have . This wavelength can be found for any particle, not just photons, and is called the de Broglie wavelength (see here for details on the historical context and experimental support for this).

Now, let us consider the corresponding angular wavenumber . Then we have

or .

The two equations, and , are known as the de Broglie relations, and the latter one will be important in the later parts of this series.