The average distance between two particles in a 1d box

[pahtakus]

Homework Statement

(i) Write an expression for the wave function of 2 particles in a 1D box. Assume that
their masses are the same, m, and that the box size is a = 1. What are the energy
states? (Comment: The particles are distinguishable)
(ii) Write an expression for the average distance between the particles hdi, quantum
mechanically.

Homework Equations

Wave function = 2sin(n1x1pi)sin(n2x2pi) for part 1 and energy levels are ((n1)^2 +(n2)^2))* h^2/4m

The Attempt at a Solution

I use <d> as an operator, and i try to operate it on the function but i don't know how to operate <d>, it wasnt taught in class and i can't find it in the textbook. is d =x2-x1. How can i find the average distance between two non interacting particles? there are no restrictions to the distance at all. They can be on top of each other? how to do it, i can't show anymore work because i don't know what to do lol.

 

[TSny]

Hello, pahtakus. Welcome to PF!

Unless I'm overlooking something, I think your expression for the energy levels is off by a factor of 2.

My interpretation of distance between the particles would be the absolute value of x₂ - x₁. So, your distance operator would be d = |x₂-x₁|. The average distance between the particles would then be the expectation value of d using your wavefunction. It's not clear from the statement of the problem if you just need to write the integral representing the expectation value or if you must go further and evaluate the integral.

The only restriction on the locations of the particles is that they must remain in the box. Yes, the particles can be on top of each other!