Combining Hamiltonians for a Two-Particle System

H_{combined}\psi_{combined}=E_{n}\psi_{combined}In summary, to find a wavefunction representing a two-particle system with identical Hamiltonians, the combined Hamiltonian should be the sum of the individual Hamiltonians. The wave function can be a product of individual wave functions or a sum of such products with arbitrary coefficients.
  • #1
erkokite
39
0
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.
 
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  • #2
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.
 
  • #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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