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Why Vcetor Spaces?

  1. Mar 19, 2012 #1
    Just a quick question here;
    Why is it that vector spaces and linear operators became the playground for quantum mechanics?
    Most of the reasons I can think of seem to come out of things derived from the fact that we're using vector spaces..
    Furthermore, who's idea was it to starting using vector spaces?
  2. jcsd
  3. Mar 19, 2012 #2
    The state of a system in quantum mechanics is a ket |ψ> which lives in a specific type of Vector space called Hilbert space. (Hilbert thought up Hilbert spaces [derr] but I think it might have been Neumann or Wigner who introduced them to physics--then again, I suspect Hilbert of jumping the gun on physics math but not admitting he did it to address physical problems. Remember he derived the Einstein Field Equations before Einstein.)

    The reason we use Hilbert space is because Hilbert spaces come with the kind of norm that is natural for a wavefunction--the L2 norm. (This was invented by Lebesgue.) The L2 norm, in mathematics, is given by ∫ψ*(x)ψ(x)dx, which as you can see is a basis representation of the quantum physics expression <ψ|ψ>. The L2 norm can be loosely described as "euclidean," and this can be interpreted as reflecting the fact that QM operates on a background of euclidean space. (Not spacetime.)

    Your question is definitely a deep one, and I encourage you to research the "Statistical Interpretation" of quantum mechanics. The question is entwined with the mysterious facts that a wavefunction is the complex root of a probability distribution, that measurable quantities are operators which act on kets, and that expectation values of measurements are given by <ψ|Q|ψ> = ∫ψ*(y)Qyψ(y)dy, (where Qy is the operator Q's expression in the y-basis. [I picked y to avoid x, which usually implies the position basis.])
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