Basics of Quantum Theory?

[Aubs]

Hello, whenever I come across bits and pieces of quantum theory I find it extremely interesting, but I have yet to come across a simplified (but not over-simplified), unified run-down of the primary components of quantum theory. I understand that the subject might not lend itself to such a thing, but if somebody would care to summarize the main points of quantum theory, that would be much appreciated. I'd like to be able to understand and take part in discussions concerning the subject.

 

As for background knowledge, I'm currently enrolled in AP Physics, understand the basics of the Heisenberg Uncertainty Principle and wave/particle duality, and have a fair knowledge of Calculus up to finding volumes of solids of revolution.

 

Thanks in advance!

Edited December 23, 2011 by Aubs

[immortal]

The Beginners guide to Quantum Mechanics by Alastair I.M Rae and Modern Physics by Moyer, Moses and Serway are some of the good books to start with the basics of QM.

 

 

[ajb]

At the fundamental level nonrelativistic quantum mechanics is all about representations of the Canonical Commutation Relations, or CCR

 

[math][x,p_{x}] = x \cdot p_{x} - p_{x} \cdot x = i \hbar[/math],

 

between the position [math]x[/math] and the associated canonical momenta [math]p_{x}[/math].

 

In particular quantum mechanics states that position and momentum are representable as self-adjoint operators on some Hilbert space.

 

The key thing this here is that physical observables become operators that in general will nor commute amongst themselves.

[DrRocket]

At the fundamental level nonrelativistic quantum mechanics is all about representations of the Canonical Commutation Relations, or CCR

 

[math][x,p_{x}] = x \cdot p_{x} - p_{x} \cdot x = i \hbar[/math],

 

between the position [math]x[/math] and the associated canonical momenta [math]p_{x}[/math].

 

In particular quantum mechanics states that position and momentum are representable as self-adjoint operators on some Hilbert space.

 

The key thing this here is that physical observables become operators that in general will nor commute amongst themselves.

 

The classification of such operators is the Stone-Von Neuman theorem which shows that they arise as irreducible unitary representations of the Heisenberg group. So one then finds interest in the representations of nilpotent Lie groups.

 

A good book on that subject is Pukanzsky's Lecons sur les representations des groupe (if one can read simple French)

[ajb]

The classification of such operators is the Stone-Von Neuman theorem which shows that they arise as irreducible unitary representations of the Heisenberg group.

 

Yes, and this theorem is important in physics as it basically tells us that Schrödinger's wave mechanics is the same as Heisenberg's matrix mechanics.