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

[Newtons Balls]

Homework Statement

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.

Homework Equations

Umm there are rather a lot :)

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 $$\sqrt{\frac{\pi}{a^3}}$$

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 a lot 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 a lot 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...

 

[Jhero]

so I'm not sure its correct.

Any help would be greatly appreciated!

Dear Scientist,

Thank you for sharing your progress on solving this problem. It seems like you have made some good progress, but I can understand your concerns about the result not seeming very profound.

One possible reason for this could be that the given potential W\delta(x-x') is a point interaction, which means that it only affects the particles when they are at the exact same position. This could result in a small interaction between the particles, leading to a small exchange splitting.

To confirm your solution, I would recommend checking it against any known results or equations for exchange splitting in a one-dimensional harmonic oscillator potential. You could also try varying the value of W and see how it affects the exchange splitting to get a better understanding of the behavior of the system.

I hope this helps and good luck with your further analysis!