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Pieced together from ancient legends by: The Thinker

**The author writes:**

You'd think that with all the training the Professor's supposedly had he'd know how to not only solve the Schrödinger Equation (Time Dependent and Time Independent), but that he'd also know how to normalise the wave function and find the expectation values for *x*† within a certain area, not to mention that he really ought to recognise the Infinite and Semi-Infinite Square Well potentials too. Instead what we have here is a complete failure at what should be relatively simple integral calculus and basic Quantum Mechanics (at least for someone of his supposed qualifications), also noting the fact that's actually Pythagoras' theorem he's got written down there- and even then he's gotten that wrong too. Turns out he's not even for real, but that somehow he received his doctorate on the back of a cereal packet; they'll print anything on those things these days.

† For one method that I've used in the past, try integrating the product of *x* with *ψ* and its complex conjugate over whatever the given boundary conditions are, whereby in the event that they're, let's say, over the range of 0 to ∞, rearrange to fit the standard definite integral of the product of *x* to the *n*th power and *e* to the negative *x* over the aforementioned boundary conditions. Like thus (for example):

<*x*> = ∫(0 to ∞) *x* *ψψ** *dx* -(rearrange to fit into)-> ∫(0 to ∞) *y*^{n} *e*^{-y} *dy* = *n*!

N.B. You might want to try something else for different boundary conditions, but this works for this particular range. What the "Professor" did came not even close to this sort of solution.