Identical, interacting spin 1/2 particles

[sevensixtwo]

Homework Statement

Given two identical, interacting spin 1/2 particles in 1D the Hamiltonian is:

H = (p1)^2/2m + (p2)^2/2m + (mw/2)*(x1-x2)^2(a) Determine the energy spectrum.
(b) Determine the corresponding eigenfunctions including spin state.

The Attempt at a Solution

I feel like I've made decent progress here. Following the method in Griffiths' QM Problem 5.1, I've redefined variables, and then used separation of variables to write two Schro Eqns. One describes the center of mass of moving freely in space and another describes the relative harmonic quantization.

According to Griffiths, the total E is the sum of the eigenvalues of these two equations. Then to answer part (a) I've said that the energy spectrum contains all finite energies greater than .5(hbar)(omega). I believe this is correct since the total two particle system moves freely in space and that aspect of the total E should not be quantized. Then I just add one continuum of energy which can be zero or greater to the (n + .5)(hbar)(omega) from the quantized part. Does anyone agree/disagree with me?

Now, for part (b) I am a little confused. Following from my separation of variables solution, the total wave function is just the product of the solutions to each Schro Eq. I have the harmonic oscillator functions. What is the wave function of a free particle? Is it just a Dirac delta?
Also, I am trying to work through Griffiths to figure out how to incorporate the spin into the total energy eigenfunctions. Any input here would really help a lot.

Thanks! Greatly appreciative!

 

[theUndergrad]

The eigenstates need to take the spin characteristic into account and will be of the form (harmonic oscillator)*(spin state). Even without the harmonic potential these particles wouldn't be truly "free" because they are constrained by the Pauli Exclusion Principle.

Typically spin eigenvalues are proportional to the inner products of the spin states, though I forget exactly...

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theUndergrad

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