Matching wave functions on a circle

Question

Let's say I have some nonzero potential in a circle with radius $R$ around zero. Just a circular potential well basically. I am stuck with a general question on matching outside and inside solutions, i.e. $\psi_{out}$ and $\psi_{in}$. Using polar coordinates, we have $\psi(r,\varphi)$. Now, I can easily match the radial part at $r=R$, but the angular part... I will have to match it for every possible $\varphi \in [0,2\pi)$, which are infinitely many. My professor wrote down, that the matching condition is thus: $$\int d\varphi e^{ik\varphi} \psi_{out} = \int d\varphi e^{ik\varphi} \psi_{in}$$ which has to be fulfilled for all $k$

In short, I dont understand this condition.

That is, I cant connect this formula to my geometrical understanding. Integrating over $\varphi$ seems reasonable to cover every point on the circle but still... I know this is the fourier transform, I suppose $k$ must be integer, which would then correspond to a series expansion??? I would be very pleased if someone could explain what is going on here.

Answer 1

The point is that for fixed $r$, any function $f(r,\varphi)$ can be expanded in Fourier modes on the circle:

$$f(r,\varphi) = \sum_{k=-\infty}^\infty f_k(r) e^{i k \varphi}$$

where

$$f_k(r) \propto \int_0^{2\pi} f(r,\varphi) e^{-ik\varphi}\,.$$ Suppose that you want to show that two functions $f(r,\varphi)$ and $g(r,\varphi)$ are identical on the circle, at $r=R$. If you define $h(r,\varphi) \equiv f(r,\varphi) - g(r,\varphi)$ this means that you must show that $h(R,\varphi) = 0$ for all $\varphi$. By the above logic, this is equivalent to imposing that $h_k(R) = 0$ for all $k$, i.e. $$\int_0^{2\pi} h(R,\varphi) e^{-ik \varphi} = 0$$ for all integers $k$. In a different notation, this is the matching condition that your professor wrote down.