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Multiple particle system

  1. Jul 5, 2009 #1
    If I have two particles, each represented by an identical Hamiltonian, [tex]\hat{H}[/tex] and I want to find a wavefunction representing the two particle system, how do I do this? I've tried to create a combined Hamiltonian and find its eigenvectors like this:

    [tex]\hat{H}_{combined}=\hat{H}\otimes\hat{H}[/tex]
    [tex]\hat{H}_{combined}\psi=E_{n}\psi[/tex]

    Note- the multiplication of the Hamiltonians is the tensor or kronecker product. I don't know if this is correct however. Could someone correct me if this is incorrect?

    Many thanks.
     
  2. jcsd
  3. Jul 5, 2009 #2

    Avodyne

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    Science Advisor

    The hamiltonian should be the sum of the two hamiltonians, not the product, because the energy of the two-particle system is the sum of the energies of each particle, not the product.

    The wave function can be taken to be just a product of individual wave functions, or more generally a sum of such products with arbitrary coefficients.
     
  4. Jul 5, 2009 #3
    So it should be equivalent to do either of these:

    [tex]H_{combined}=H_{1}+H_{2}[/tex]
    [tex]H_{combined}\psi_{combined}=E_{n}\psi_{combined}[/tex]

    or

    [tex]\psi_{combined}=\psi_{1}\otimes\psi_{2}[/tex]
     
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