Boundary condition like $\psi(0)= \psi(L) = 0$

Question

This might be very elementary. But I have been baffled for a while.

For the infinite square well potential, the boundary condition is that $\psi(0) =\psi(L) = 0 $. However, from real analysis, we know that we are free to modify the value of a function on any set of measure zero. So what is the point of specifying or constraining the values of $\psi$ at the ends to zero? $\psi$ could take the constant value of 1 everywhere except at the ends.

Answer 1

Yes, elements of $L^2([0,L], dx)$ are defined up to zero-measure sets. However here we have a specific operator: the kinetic energy operator $-\frac{d^2}{dx^2}$ (in weak sense in order to be selfadjoint). The domain of the kinetic energy operator is the second Sobolev space and, in one dimension, the elements of that space are continuous (each equivalence class has a continuous element).