Two particles in QHO potential and probability of distance

[jmm5872]

Two particles with equal mass m are confined by a harmonic oscillator potential
of frequency ω. Assume that we somehow know that there is one particle in the first excited
state and one in the second excited state. What is the probability of finding both particles
within a certain distance if they are noninteracting particles?

I am mostly confused on exactly how to approach the distance here.

ψ(x1,x2)=ψ₁(x1)ψ₂(x2)

To find the probability of each particle in some range we would ∫∫ψ²dx1dx2 over [x1,x1+Δx1] and [x2,x2+Δx2]. (Δx1=Δx2)

This will give me the probability that particle one is somewhere within [x1,x1+Δx1] AND particle two is somewhere within [x2,x2+Δx2].

I'm not sure what to do to find the probability of both particles within some distance. I can integrate both over that distance, but this probability will be different at different locations in the distribution function. Or, the relative distance between the two can be the same, but the probability will be different (for the same distance) at different locations in the potential. I'm not sure if that makes sense.

Probability of [1,1+Δx1] and [1,1+Δx2] will be different from the probability of [0,Δx1] and [0,Δx2] even though they are the same distance. So would I need to somehow take the product over all the probabilities?


Or maybe I am overthinking this and the question is simply asking for the probability of both particles being with [-Δx/2,Δx/2].

 

[vela]

Try using the change of variables
\begin{align*}
u &= \frac{1}{\sqrt{2}}(x_1+x_2) \\
v &= \frac{1}{\sqrt{2}}(x_1-x_2)
\end{align*}