People say that Quantum Mechanics is hard, and they're not wrong; the mathematcial formalism is heavy, and does its best to obscure the underlying Physics. Dirac's elegant formalism does a lot to clarify it, and also helps you understand the Maths if you know the Physics.

Before I continue, it should be mentioned that I'm assuming knowledge about vector spaces, bilinear forms, complex numbers, a little about hermitian inner products and basic knowledge about integration and stuff. If I had to explain all this too, this write up would be very, very long.

A brief outline of the theory so far: In Quantum Mechanics, the state of a particle can be expressed by a wave function, a complex-valued function of position, whose absolute square gives the probability distribution function for finding the particle in any given region of space. The reason for it being the absolute square and not anything simpler is that given two wave functions for different states of a particle, they can be superimposed with a resulting wave function that is the sum of the previous two. Since it is now the absolute square of this sum (and not the sum of the absolute squares) that gives the combined probability distribution, this allows wave-like interference to occur between the particles.
To calculate the probability of transition between two states with wave functions φ (x) and ψ (x), complex conjugate one, multiply it by the other and integrate over all space.
If you then want to measure some property of the particle, say its energy, you can do this by acting a suitable linear operator on the wave function; if the wave function is an eigenvector of the operator, then the measured observable (energy in this case) is the associated eigenvalue. If not, then the particle is said not to have a definite value of that observable.

Dirac's formalism treats all states of particles as element of a complex vector space, written as |ψ> and called kets. There is also a corresponding dual space of states, with elements <ψ| which are called bras. The two spaces are indentified by the standard hermitian form, sending <a| and |b> to a complex number <a|b>. Don't think about these vectors as being more-or-less the same thing as the wave functions, as I haven't yet said anything about spatial dependence.
As with the wave functions before, linear combinations of kets give the ket of the combined state, and when acting operators, which I will now specify need be hermitian, on the kets we identify the eigenvalues with the observables (viz. H, the energy operator, has eigenvalue E on the state |φ>; H|φ> = E|φ>). We also define the probability amplitude for the transition |ψ> to |φ> as <φ|ψ>.
Knowing that for any hermitian operator, we can find a complete orthonormal basis of its eigenvalues, any state |ψ> can be written uniquely as:

> = c1|1> + c2|2> + ...

where the |i>s are the eigenvectors of the operator concerned. Using the standard hermitian form, we work out that ci = <i|ψ>, which gives rise to the very natural looking expression:

> = |1>< 1|ψ> + |2>< 2|ψ> + ...

and it is this that ties in with the lemma in the proof of the Gram-Schmidt Theorem. In fact, a really neat way to write it would be:

| = |1>< 1| + |2>< 2| + ...

because there's nothing special about the ψ. But what does it mean? I said already that the probability amplitude for a transition was <φ|ψ>, and applying the above to this gives <φ|ψ> = <φ|1>< 1|ψ> + <φ|2>< 2|ψ> + ... ie. the amplitude for getting from one state to another is just a sum over all intermediate states of going from ψ to φ via the intermediate state. Quite neat.

So what about wave functions? Position is an observable, and has eigenstates |x>; then <x> is the amplitude for ψ being a state x ie. being at x - so this is just the wave function we had before. Note that this, combined with the intermediate transition rule before gives back the old rule of one wave function going to another (noting that we are now integrating over the intermediate states: <φ|ψ> = ∫ <φ|x><x> d3x .
Now of course there's nothing special about position: would could use momentum eigenstates |p> instead. Knowing the relationship between the position and momentum operators, you can show that the functions <p> are essentially the Fourier transforms of the usual wave functions, and then by applying the above relation twice you get the Fourier inversion theorem for free! (Pure mathematicians may wish to object at this point).

If you're currently trying to learn about QM, and are looking for a good book to read, it is important to find one written by someone who understands and enthuses about the formalism (comparatively few do). The best two that spring to mind are:

Ironically, Dirac's own textbook The Principles of Quantum Mechanics is not so lucid as the above two.

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