What is the meaning of this equation?

[josus10]

Thinking back to my intro to modern physics course a while back, one particular equation always sticks out to me but I never quite understood how to interpret it properly. We were doing some basic QM and looking at the normalization of the wave function in 1D. It looked something like this;

∫|Ψ(x)|²dx

If I remember correctly but I don't have my book with me. The limits of integration were from -∞ to ∞ and then were readjusted to be from 0 to L (L being the length of the box or well.) I'm hoping that I didnt make a mistake typing it but provided I didn't, what is the meaning of the infinite limits of integration? I have always been thinking of it as the definition of existence, where the infinite bounds represent the bounds of the universe, but the more I think of it that way the less sense it makes. Any thoughts?

 

[DrClaude]

The wave function is defined over all of space, therefore the integration has to be done from -∞ to ∞. Thinking of it as "the bound of the universe" is not helpful, as that runs into conceptual problems. It is in a sense a mathematical construct, but it works very well.