Welcome to Gaia! ::

Let's begin with a few definitions:

A linear operator (transformation, map, etc.) is something that takes a vector (basically a point in space) and spits out another vector, with the following properties:
Given a linear operator A, vectors u and v, and a constant c,

and

Rotations around the origin and dilations are examples of linear operations on vector spaces.

Now our vectors in the case of quantum mechanics are actually functions that take a real number and spit out a complex number. Instead of the familiar vectors which can be characterized by a finite set of coordinates, these functions have a coordinate for every point in R, this coordinate being the value of the function at that point.

Now we define the Hermitian Inner Product:
Given two functions f(x) and g(x), we define


Note that for functions f, g, and h, = + and that for a constant c, = c. Also note that = *.

For any given wavefunction
we can find some c such that

We then consider to be normalized. For convenience, unless otherwise noted we assume that all wavefunctions in question are normalized.

So what does the wavefunction mean?

Well, one way to read it is that is the probability that the object is between points a and b at time t. Note that a and b might need to be very spread out to get a significant probability.
As for the wavefunction itself, nobody's quite sure what it actually is. It might be reality, it might just be a representation of our knowledge of the situation. This is called the measurement problem, and will be returned to.

A couple things to note about observables and operators:

The operators must be what is called "Hermitian" in that given any wavefunctions f(x, t) and g(x, t), = . Why is this? This is because the constants q must be real; I'll leave the rest to you to figure out.

Similarly,


From statistics, we get that:

Now how about the momentum? Well, let's look for the change in the (average) position over time:



Since the integral is independent of time, we can move the derivative w.r.t time inside the integral, getting



Now, by the product rule, we get that



Applying Schroedinger's equation to the first term on the right hand side gives us:



And then to the second, we get the complex conjugate:



Sticking those back into the equation above gives us that



Thus



A bit of integration by parts gives us



Multiplying both sides by m gives the momentum.

Thus we get that the appropriate operator for the momentum is

A very good idea. And OMG you found an online LaTex thingy! *steals*

I suppose it's hard to know where to start regarding Quantum mechanics at this level (i.e. Beyond the 'did you know photons act strange!?') level.

One small suggestion though: I don't think you can stress the Schrodinger equation enough regarding QM.

And before even touching operators or dirac notation, I would stick with wave mechanics.

It might be a good idea to start with the most general description of a massless particle in 1-dimension:



You can write it explicitly as:


or, more usefully:


For your purposes, you'd only need to explain that all the information regarding the dynamics of this free particle is encoded in the above function. It wouldn't be necessary to delve into fourier superpositions and whatnot ninja .

And then explain that, to undestand how this particle evolves with time, we must construct a time evolution equation, an equation that tells us how this particle will evolve with time. The Quantum mechanical equivalent of:



You do this first by calculating some derivatives of .

Namely:



and



We know that


so after a little substitution of E and p^2 we get:



Tadaa! The Schrodinger equation for a free particle. heck, with a little extra you could throw in the Klein-Gordon equation.



If you like we could collaborate and write up a sticky thread for this guild regarding QM. cheese_whine

I suppose if we were going to make a sticky about QM we would have to decide exactly who we're aiming it at. And decide what we could assume, as opposed to explicitly derive.

Whenever I give QM introduction talks at my university, I generally start with the wavefunction of a massless free particle and build from there by outlining its properties and by demonstrating the correspondence principle. The conjugate nature can be explained with fourier transforms and canoncial commutation relations.

But it's much of a muchness really.

Hmmm... Well what I could do is write up a heuristic development of the wave equation and give some background on wavefunctions. That would make any mathematical exploration you undertake a lot less 'intimidating' to the average gaian.