Homework Help: Atomic Physics, two particle system in 1-d harmonic oscillator

1. Mar 12, 2008

Newtons Balls

1. The problem statement, all variables and given/known data

Two identical spin 1/2 particles of mass m exist in a one dimensional harmonic oscillator potential $$\frac{kx^{2}}{2}$$ where x is the position coordinate and k is a constant. The particles interact with eact other with a potential W$$\delta$$(x-x'), where $$\delta$$(x-x') is a Diract delta function of the particle coordinates x and x', and W is a constant. Obtain the exchange splitting of the lowest excited energy level of the system for small W by considering the states to be the triplet and singlet configurations when the particles share the two lowest one-particle spatial states of the system.

2. Relevant equations
Umm there are rather alot :)

Schrodinger's Equation gives a nasty solution containing a Hermite polynomial... I don't want to type these two into latex.

It gives a known integral of between infinity and minus infinity:
$$\int$$y$$^{2}$$exp(-ay$$^{2}$$)dy = 1/2 $$\sqrt{\frac{\pi}{a^3}}$$

3. The attempt at a solution

Well at the moment we're strugging to even start, we can get the wave functions of the individual particles but combining the two into a joint wavefunction is tricky. After that I believe you integrate that multiplied by the given potential which should be in the form of the given integral. The problem is finding the wave function consisting of both the particles.

I don't really need alot of general help on this, just any advice on how to combine the two wavefunctions of the individual particles into one consisting of both. If you want me to type up the solution for each particle I can but it'd take alot of time with latex.

Hopefully this makes some sense... we've been looking at this for a few hours with quite limited progress

Thanks!

Update: I think I've managed to solve this in the end but have doubts.

It resulted in a solution of W(mk)$$^{1/4}$$/(2h)$$^{1/2}$$

Which is small(?) when W is small, so perhaps I made a mistake as that doesn't seem very profound.

Solution found by using standard wavefunction result for particle in a harmonic oscillator which is a product of hermite polynomials, combined the two by using a slater determinant. Then used an equation I dug up which indicates that the exchange splitting is 2X where X is:
$$\int$$phi0(x).phi1(x').phi1(x).phi0(x').P dx dx'
Where P is the potential between the two particles, phi0 and phi1 referring to the wave function in the ground and second state respectively.

Using the given values for the potential and the two wave functions it all came out nicely and was in a form which matched the solution to the given integral which indicates its in the right direction.

It all comes out nicely to the result stated above, but I'm a bit worried about the 'for small W' mentioned in the question doesn't seem to do anything useful...

Last edited: Mar 12, 2008