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I study mathematics and am in my sixth year, but...

I have a very elementary question:

I stumbled upon it while learning for quantum mechanics. But it's nothing new, it's happening to me all the time: I get confused by things like this!

Observe the following facts:

Suppose we deal with the space of wave functions over the real line.

The wave function [tex]\psi(x)[/tex] is a complex scalar. Take for example [tex]\psi(x)=e^{ikx}[/tex] (not normalizable, don't need it)

The derivative [tex]\psi'(x)=ike^{ikx}[/tex] has 1/length as it's unit.

Integrating that over some interval yields a scalar. [tex]\psi(b)-\psi(a)[/tex]

The probability density [tex]\psi*(x)\psi(x)[/tex] is a scalar.

Integrating this over the real line (length) gives 1. A scalar...

Shouldn't such an operation yield a length? Am I stupid?

I am not joking. For me this is a mystery.

Thanks for your answers.

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# Very stupid question

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