**Required math: calculus **

**Required physics: **Schrödinger equation

Reference: Griffiths, David

J. (2005), Introduction to Quantum

Mechanics, 2nd Edition; Pearson Education – Chapter 2, Post 20.

While analyzing the free particle, we saw that we could construct a normalizable combination of stationary states by writing

We can find the function by specifying the initial wave function:

This relation can be inverted by using Plancherel’s theorem, which states

Here we run through a plausibility argument which is a sort of physicist’s proof of Plancherel’s theorem. We start with Dirichlet’s theorem which says that any (physically realistic, anyway) function can be written as a Fourier series. We can show that this is equivalent to a series in complex exponentials. That is

We’ve used the facts that cosine is even and sine is odd. This is equivalent to a Fourier series:

where the coefficients are related by

Inverting the relations we get, for

We can get the coefficients in terms of by integration:

The integral is zero if and if , so the right hand side comes out to just and we get

Now we can make the substitutions

If is the increment in from one to the next, then . We can then write the original series as

The formula for now becomes

Now we can take the limit as . In this case, (that is, it becomes a differential) and the sum becomes an integral, so we get

In the second formula, the limits on the integral become infinite, and we get the other half of Plancherel’s theorem:

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Ziruo ZhangHi! I was just wondering if there should be a third case apart from equations (11) and (12) which specifies that c0=b0? Otherwise how should we deal with the case where n=0? Thank you very much! Your answers to these problems have been really helpful!

gwrowePost authorI have already specified that in equation 8. Equation 14 works for as well, as it just says that is the average of over the interval .