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Well if we're actually serious about making an official 'How to' guide for the physics and Mathematics guild then sure, the more the merrier. Only thing is before actually starting this guide perhaps we should send draft copies to one another for proof reading and whatnot, as if we each write individual sections without conferring then ya get a disjointed fog.

Ok, how about these for the postulates:



Postulate 1:

The quantum state of a system at any moment is completely characterised by the wavefunction.



This function is not a physical observable.

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Postulate 2:

The wavefunction of a physical system evolves according to the schrodinger equation



Where is the Hamiltonian operator for the total energy



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Postulate 3:

For any measurable physical quantity, which is a function of position and momentum co-ordinates, there is a correspongind operator .

This operator is Hermitian and is called an observable.

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Postuate 4:

The only possible results from measuring a physical quantity, , is one of the eigenvalues associated with it. Each eigenvalue has an eigenfunction associated with it. The probability of obtaining the eigenvalue is



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Postulate 5:

If the measurement of a physical quantity on the system gives the result , then the state of the system immediately after the measurement is


Maybe we could explain each postulate in detail.

Actually, talking about Hilbert space would be a very good idea, because then we could explain that wavefunctions with completely defined position or momentum don't exist physically because they aren't in the Hilbert space (I kept forgetting why position eigenfunctions aren't in the Hilbert space; it took me a while to remember that the Dirac delta isn't technically a function).

Yeah, we should definitely give each postulate a separate post to explain it, as none of them match up to the classical postulates. For example, we might want to explain why the wavefunction isn't observable.


I think the material should go something along the lines of:

Mathematical prereqs:

-Complex numbers, conjugation, and modulus
-Multivariable notation, assuming knowledge of single variable calculus
-Vector spaces, linear operators, non-commutation, maybe inner products. Eigenvalues and eigenvectors.

(with a note along the lines of "if you are familiar with these, skip to...")

Postulates of Quantum mechanics

-Wavefunctions
-S.E.
-Observables and operators
-Observation
-Superposition collapse

(each postulate gets a separate post)

Results of the postulates

-Ehrenfest's theorem
-Quantization of certain observables
-HUP+general uncertainty
-Tunneling

Other stuff

-The meaning of the wavefunction
-The measurement problem

Hmm.. How about:

Since the result of an observation must be real, this means the eigenvalues associated with the operators must be real. This can be expressed mathematically as



This is possible if the operators (observables) are "hermitian". Hermitian, for the purposes of this introduction, simply means the following condition is satisfied:



Where and are functions which satisfy certain boundary conditions.


It can be easily shown that the eigenvalues of such Operators are real. Consider , an eigenfunction of with eigenvalue :







Divide both sides by



and you have




Tadaa!

I want to reply more thoroughly later, but I'd like to note the Word of the Physics gods...err, I mean, what Shankar writes as the comparison between classical mech and QM:

1. In classical mechanics, the state of a particle at any given time is completely specified by the two variables x(t) and p(t), i.e. as a point in phase space. In quantum mechanics, the state of a particle is represented by a vector |psi(t)> in a Hilbert space.

2. In classical mech, every dynamical variable omega is a function of x and p. In QM, x and p are represented by Hermitian operators X and P with the following matrix elements in the eigenbasis of X:
= x delta(x-x') and = -ihbar delta'(x-x')
The operators corresponding to the omega(x,p) are Omega(X,P) = omega(x ->X, p -> P).

3. In CM, if the particle is in a state given by x and p, the measurement of the variable omega will yield a value omega(x,p) and the state (x,p) will be unaffected. In QM, if the particle is in a state |psi>, measurement of the variable (corresponding to) Omega will yield one of the eigenvalues omega with probability P(omega) proportional to ||^2. The state of the system will change from |psi> to |omega> as a result of this measurement.

4. In CM, Hamilton's equations. In QM, Schrodinger's equation.

Shankar builds up everything else in elementary (one-particle) QM from these, so I know it's at least possible. They also appear to be pretty minimal hypotheses for describing the quantum world (though they're not the best tools for understanding it).

Oh, S.E. is Schroedinger's equation. I just didn't want to have to write that a billion times, plus that's what Griffiths, the textbook I work out of, abbreviates it as. In fact, most of the literature I find abbreviates it as S.E. for convenience.

I'm also going to be busy for the next few weeks. My summer tutorial is winding down, and with the end comes the deadlines for the papers. And then I have to read about a hundred dense-as-hell pages for my research, and my dad wants me to start on several programming books.

Due to the way Harvard works, I'll actually have more time when school starts, since we have a "shopping period" for classes, i.e. the fun stuff doesn't start for a week. In the mean time, I'm cramming as much as I can into summer research/independent reading, because as soon as classes start the professor I'm working with becomes really busy.

Well we're not gonna be writing anything too comprehensive I assume. Just some insight into the workings behind quantum mechanics. Basically to stop people abusing cats.

I'd recommend posting it up on the science forum as well when we're finished.


And yay for holidays rofl