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System of 2 particles: why is the wavefunction a product?

  1. Feb 6, 2015 #1
    I am trying to solve for the energy of 2 non-interacting identical particles in a 1D infinite potential well. I want to do it as much "from scratch" as possible, making sure I fully understand every step.

    H = -ħ2/2m * (∂2/∂x12 + ∂2/∂x22)

    Hψ=Eψ

    2ψ/∂x12 + ∂2ψ/∂x22 = kψ, where k=-2mE/ħ2

    I got up to here, where I need to write down a form for ψ to take. Both textbooks I referred to (Griffiths and Shankar) use the example of a wavefunction for 2 distinguishable particles before discussing the 2-term form for a wavefunction of 2 indistinguishable particles. Apparently for 2 distinguishable particles, ψ(x1,x2)=ψa(x1b(x2), but I don't understand the reasoning behind writing it as a product of the individual wavefunctions. Griffiths discusses it a little bit in footnote 2, but I don't follow it. I do understand the later part about writing ψ for indistinguishable particles as a sum of the two products, but I don't understand why each term is written as a product in the first place.
     
  2. jcsd
  3. Feb 6, 2015 #2

    atyy

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    It can be taken as an additional axiom that the Hilbert space for two particles is the tensor product of the individual Hilbert spaces, eg. http://www.theory.caltech.edu/~preskill/ph219/chap2_13.pdf (Axiom 5). Then the basis functions of the two-particle Hilbert space are products of basis functions of one-particle Hilbert space.
     
  4. Feb 6, 2015 #3

    bhobba

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    Atty is correct.

    Its an axiom, but its so natural and obvious many textbooks don't state it.

    Thanks
    Bill
     
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