- #1
snatchingthepi
- 148
- 38
- Homework Statement
- Consider the symmetry and antisymmetric two-particle wave functions for a one-dimensional box with impenetrable walls at x = +- L/2. One particle occupies the ground state, and the other occupies the first excited state.
What is the probability to find a particle at position x for either case if we do not care about the position of the second particle.
- Relevant Equations
- See below
So for the 1D infinite well with the states above, I have
## \psi_{symmetric} = \frac{2}{L} [sin[\frac{\pi x_1}{L}]sin[\frac{2\pi x_2}{L}] + sin[\frac{2\pi x_1}{L}]sin[\frac{\pi x_2}{L}]] ##
## \psi_{antisymmetric} = \frac{2}{L} [sin[\frac{\pi x_1}{L}]sin[\frac{2\pi x_2}{L}] - sin[\frac{2\pi x_1}{L}]sin[\frac{\pi x_2}{L}]]##
The question statement says to find the probability of finding a particle at a position ##x## for both cases if we "do not care about the position of the second particle". How do I do that? I thinking I might be able to simply integrate over the whole range for one particle, and then integrate from one edge of the well to the position x for the other? But I've never done anything like this and do not know.
## \psi_{symmetric} = \frac{2}{L} [sin[\frac{\pi x_1}{L}]sin[\frac{2\pi x_2}{L}] + sin[\frac{2\pi x_1}{L}]sin[\frac{\pi x_2}{L}]] ##
## \psi_{antisymmetric} = \frac{2}{L} [sin[\frac{\pi x_1}{L}]sin[\frac{2\pi x_2}{L}] - sin[\frac{2\pi x_1}{L}]sin[\frac{\pi x_2}{L}]]##
The question statement says to find the probability of finding a particle at a position ##x## for both cases if we "do not care about the position of the second particle". How do I do that? I thinking I might be able to simply integrate over the whole range for one particle, and then integrate from one edge of the well to the position x for the other? But I've never done anything like this and do not know.