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I was told in class that a wave function should have the following properties:

  1. Finite and single-valued
  2. Continuous
  3. Differentiable
  4. Square integrable

But if we consider the wave function in an infinite square well, the wave function isn't differentiable at the boundaries since $\Psi (x)$ is:

$$\Psi (x) =\begin{cases} \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}), & 0<x<L \\ 0, & \text{elsewhere} \end{cases}$$

This violates one of the properties of wave functions. So how is this an acceptable wave function?

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Like with a plane wave, the wave function you are giving is only a "limiting case". In reality you should start with finitely high walls and impose all four requirements that you list. Then, as you take the limit of higher and higher walls, you will obtain your wave function. However, think of it only as a limiting case. In reality, walls will always have finite height.

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