Quantum Mechanics

Hi, I've started to watch some lectures on quantum mechanics & they're going well except for the fact that some of it makes no sense. Basically I just don't see how |ψ|²dτ represents the probability of finding a particle described by ψ in the volume element dτ. Most likely it's due to me having missed something in the development of the material thus far or maybe it's because the answer to this question hasn't been fully given yet - or maybe I'm just missing some elementary logic. I think it's best to briefly state exactly what's been done in the course:

History
Defined Schrodinger Equation
Mentioned, without justification, the Born Interpretation:
- The element |ψ|²dτ represents the probability of finding the particle described by ψ in the volume element dτ
Derives the 1-Dimensional Schrodinger equation & momentum operators
Evaluates some Gaussian integrals
Gives four representations of the dirac delta function
- via a Rectangle function
- via the Heaviside step function
- via a Gaussian function
- via the sinc function (using this to give an integral representation)
Goes through Fourier's integral theorem
Derives Parseval's identity (Also known as the normalization condition)
Inserts the Debroglie relation into the integral representation of the Dirac delta function & uses Parseval's identity to derive a relationship between position & momentum
Derives a solution to the 1-D Schrodinger equation for a free particle
Shows how the equation of continuity can be derived from the Schrodinger equation
Mentioned the commutator [a,b] = ab - ba
Just started to derive material related to expected values
(If any of this is too short I'll expand on it)

So I can see how normalizing the integral ∫|ψ|²dτ due to the linearity of the Schrodinger equation is a way to get ∫|ψ|²dτ = 1 & that this obviously relates things to probability but I mean to make such a definitive claim about what |ψ|²dτ stands for seems a bit much.

Just a thought , if anyone thinks worthy of a response.


Is there any merit, do you think, in pursuing the possibility of electrical power generation via the interaction of the earth's magnetic field together with the moving conductor (viz sea water) , on principle?

(I have not done, nor do I have the ability to, any analysis on the energy yield)

I have unstuck this thread for now, as I'm not sure how useful it is as a reference.

But I am open to suggestions.