Rayleigh Ritz with two Particles

  • Thread starter wduff
  • Start date
Hello everyone, I have what should be a simple one to answer.

I'm solving for 2 particles in a harmonic oscillator with a gaussian bump in the middle and a delta function interaction. I'm doing all this via Rayleigh Ritz; that is, diagonalizing the Hamiltonian to find the constants in:

[tex]\Psi = \sum_{ij} c_{ij} \psi_{ij}[/tex]

Where ##\Psi_{ij}## is just the standard symmetrized 2 boson wavefunction with ##\psi_{i}## and ##\psi_{j}## in a harmonic oscillator basis.

My issue is this: I end up with a rank 4 tensor: [tex]\langle \psi_{ij}|H|\psi_{kl}\rangle[/tex]

I'm sure this is just inexperience, but I have no idea how to get an eigensystem out of that (I'm using mathematica). I've already done it with one particle (minus the delta function of course), which was simple enough, since the calculations produced a n x n matrix which I know how to handle. But this is a new one for me.

Any suggestions would be very appreciated. Thanks!
 
Last edited by a moderator:

DrClaude

Mentor
6,807
2,924
One needs to define a basis of two-particle states, which are themselves the tenor product of single-particle states.

In that two-particle basis, the Hamiltonian will be a 2x2 matrix.
 

Want to reply to this thread?

"Rayleigh Ritz with two Particles" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top