Equal or larger/smaller versus larger/smaller in boundary conditions

[Tokki]

TL;DR Summary

I am confused about why the two seem to be approached the same in all the solutions I have seen

Hi everyone!
This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent!
I am working on infinite square well problems, and in the example problem:
V(x) = 0 if: 0 ≤ x ≤ a
∞ : otherwise

H`ere it is obvious that one should take ψ(a)=ψ(0)=0 and solve.

In the next example problem, however, the boundaries are shown as follows:
V(x) = 0 if: -a < x < a
∞ : otherwise

Here potential is zero if x is between (-a) and (a) but not when x is equal to (-a) or (a).
However, in the solution, the author does the same thing, making ψ(-a)=ψ(a)=0
The conditions, however, do not seem to allow this.
What am I missing?
Thank you!

 

[PeroK]

Summary:: I am confused about why the two seem to be approached the same in all the solutions I have seen

Hi everyone!
This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent!
I am working on infinite square well problems, and in the example problem:
V(x) = 0 if: 0 ≤ x ≤ a
∞ : otherwise

H`ere it is obvious that one should take ψ(a)=ψ(0)=0 and solve.

In the next example problem, however, the boundaries are shown as follows:
V(x) = 0 if: -a < x < a
∞ : otherwise

Here potential is zero if x is between (-a) and (a) but not when x is equal to (-a) or (a).
However, in the solution, the author does the same thing, making ψ(-a)=ψ(a)=0
The conditions, however, do not seem to allow this.
What am I missing?
Thank you!

The two problems are physically the same, whether you have $<$ or $\le$.

Why do you think in the second case you don't have $\psi(-a) = \psi(a) = 0$? It can't be non-zero at these points.