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 and let our “state vector”

Now consider a (bogus) state operator which has an eigenbasis and with eigenvalues (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 as required for an orthonormal basis with this inner product.

Now we can write

(yes, I am abusing the equal sign here)

This means that

Now the only possible measurements of the operator are 0, -1, -4, -9, …. and the probability density function is:

One can check that

Here is a plot of the state function (blue line at the top) along with some of the eigenfunctions multiplied by their respective .

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