# College Math Teaching

## July 15, 2011

### Quantum Mechanics and Undergraduate Mathematics III: an example of a state function

I feel bad that I haven’t given a demonstrative example, so I’ll “cheat” a bit and give one:

For the purposes of this example, we’ll set our Hilbert space to the the square integrable piecewise smooth functions on $[-\pi, \pi]$ and let our “state vector” $\psi(x) =\left\{ \begin{array}{c}1/\sqrt{\pi}, 0 < x \leq \pi \\ 0,-\pi \leq x \leq 0 \end{array}\right.$

Now consider a (bogus) state operator $d^2/dx^2$ which has an eigenbasis $(1/\sqrt{\pi})cos(kx), (1/\sqrt{\pi})sin(kx), k \in {, 1, 2, 3,...}$ and $1/\sqrt{2\pi}$ with eigenvalues $0, -1, -4, -9,......$ (note: I know that this is a degenerate case in which some eigenvalues share two eigenfunctions).

Note also that the eigenfunctions are almost the functions used in the usual Fourier expansion; the difference is that I have scaled the functions so that $\int^{\pi}_{-\pi} (sin(kx)/\sqrt{\pi})^2 dx = 1$ as required for an orthonormal basis with this inner product.

Now we can write $\psi = 1/(2 \sqrt{\pi}) + 4/(\pi^{3/2})(sin(x) + (1/3)sin(3x) + (1/5)sin(5x) +..)$
(yes, I am abusing the equal sign here)
This means that $b_0 = 1/\sqrt{2}, b_k = 2/(k \pi), k \in {1,3,5,7...}$

Now the only possible measurements of the operator are 0, -1, -4, -9, …. and the probability density function is: $p(A = 0) = 1/2, P(A = -1) = 4/(\pi^2), P(A = -3) = 4/(9 \pi^2),...P(A = -(2k-1))= 4/(((2k-1)\pi)^2)..$

One can check that $1/2 + (4/(\pi^2))(1 + 1/9 + 1/25 + 1/49 + 1/81....) = 1.$ Here is a plot of the state function (blue line at the top) along with some of the eigenfunctions multiplied by their respective $b_k$.