Combining Hamiltonians for a Two-Particle System

[erkokite]

If I have two particles, each represented by an identical Hamiltonian, \hat{H} 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:

\hat{H}_{combined}=\hat{H}\otimes\hat{H}
\hat{H}_{combined}\psi=E_{n}\psi

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.

 

[Avodyne]

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.

 

[erkokite]

So it should be equivalent to do either of these:

H_{combined}=H_{1}+H_{2}
H_{combined}\psi_{combined}=E_{n}\psi_{combined}

or

\psi_{combined}=\psi_{1}\otimes\psi_{2}