Re: Quantum physics study - Where to begin?

Gregory L. Hansen wrote:
Although introductory texts rarely say much about this, the big change from classical to quantum physics is the representation of state going from a vector in a phase space to a vector in a Hilbert space. That step does away with determinism and defines the wavefunction and the interpretation of it.

I don't agree. That step can be factored into two smaller steps. In the first half-step, you replace the vector in a phase space with a classical probability distribution over vectors in the phase space. The set of unnormalized distributions of this kind forms a Hilbert space whose dimension is equal to the number of points in the original phase space. The equivalence classes of such distributions, modulo normalization, are lines through the origin. Determinism is lost, but the predictions of the theory don't change; it's still classical.

In the second half-step you simply replace the real probabilities with complex amplitudes. That's where things become quantum. I think those amplitudes are what quantum mechanics are really about. Some formulations of quantum mechanics use a Hilbert space and others don't. Some have a collapse postulate and others don't. Some quantum theories use particles and others use waves. But the complex amplitudes are always there.

You don't have to be able
to actually find the hydrogen energy eigenfunctions given naught but a pencil and blank paper before you can claim some understanding of what quantum mechanics is all about. But you do have to know what an eigenfunction is.

Well, it's a good idea to know what an eigenfunction is, but you certainly don't have to. Eigenfunctions are peculiar to the operator formalism. If there's no degeneracy, you can simply specify an orthonormal basis for the Hilbert space and a measurement outcome for each basis vector. It's only when you have degeneracy that the operator-eigenvector formalism becomes useful, and even then it's not really necessary, just more elegant (since it preserves the symmetry associated with the degeneracy).

But the textbooks are filled with differential equations, wave mechanics, Bessel functions, and so on. It's too bad, really, because I think the student is forced to wade through a lot of material before he gets to what is essential about the theory. 

I agree with that.

-- Ben
.

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