Hmmmm.... It would seem that everyone is either getting nowhere or is too scared to even try it. Well, that makes me feel a bit better about not getting it myself!
I've been trying this one by the "brute force" method and have gotten essentially nowhere, so I was wondering if there is a more elegant way to approach it.
A particle is in a 1-D infinite potential well. It is a free particle over the interval [0,L]. Presume that we know the particle is in a state such that we know it is at position L/2 with certainty.
Given the structure of the potential we can expand this wavefunction using the orthonormal basis states: where n spans the positive integers. So:
where the are complex in general.
The x=L/2 supposition gives us the following conditions for all integer k:
From basic Quantum Mechanics we have the following condition as well:
I need to find either the coefficients , or more generally:
Now, the problem does go on to say that we can ignore such "trivial" issues such as convergence of the series, but I suspect there is only one solution for the anyway. (That could easily be wrong!) I can say, with certainty, that there are an infinite number of non-zero coefficients, so the series doesn't terminate (making the problem simple, if long.) I suspect as well that there are no non-zero coefficients. I'm pretty sure about this one.
I have tried to do this by coming up with conditions on the by working out the integrals for various values of k, but that approach doesn't appear to help much. I can't seem to separate the terms of the various infinite series such that they are of any use.
As Sherlock Holmes would say, it's quite a "three-piper!"
By the way. One of the deficiencies of my education is that I know only one type of (continuous) function that would produce a state of certain position x = L/2: a Gaussian normal centered on L/2. However we need the additional condition that the wavefunction goes to 0 at x = 0 and x = L, which a Gaussian won't do. Perhaps a tactic for solving the problem would be to apply various functions that DO have this property and see what happens with the expansion into the orthonormal basis, but I don't know how to construct such functions in general.
Substitute into the expression , the convergence being justified by the orthogonality of the system in [0,L].
A few calculations later, we get
Hope that helps .
By the way. Your function will not be a Gaussian in the usual sence, but something quite like it, and also vanishing at the endpoints.
Thank you for the attempt. Unfortunately I already knew that. What I need to know is the actual value of the integral and as I don't know what the cn's are I can't leave the result that way.Originally Posted by Rebesques
In other words I need some clever way to evaluate the integral such that the result doesn't depend explicitly on the cn's or a way to find what the cn's are using the x^k integral conditions.
Well, the problem is that my series techniques have never been that advanced. I can work out any number of conditions on sums involving the cns using the x^k integrals, but I can't seem to find a way to use them to get an explicit expression for the cn out of them. I was hoping that someone here has that kind of expertise and guide me, or show me a clever way to get around using them. (sigh)Originally Posted by Rebesques
It's only a homework problem, and a self-imposed one at that, so no big deal if I can't find a way to solve it but my curiosity is up about this one.