# Three particles in a one-dimensional box of length L.

## Homework Statement

Three particles, each of mass m, reside in a 1D "box" of length L. Find a complete set of wavefunctions,$\phi_{k}(x)$ for the system. In this sense, "complete" means that any state of the system,$\psi(x)$ can be written as a superposition of the wavefunctions, $\phi_{k}(x)$.

## Homework Equations

The TISE and TDSE, as well as other standard QM expressions.

## The Attempt at a Solution

Well, there are no applied fields/forces, so I think I'm correct in saying that the Hamiltonian is time-independent (please correct me if this is not the case). This is a many-body problem, so I've attempted to use methods such as mean-field theory (Hartree-Fock), and second quantization, but run into trouble because the question doesn't give any clues as to which particles are being considered, or any boundary conditions.
I'm not sure if this is the most elegant way to complete the problem. Am I missing something simple? I know that each particle will be described by a separate wavefunction, and each linear combination of wavefunctions will also be a solution to the TDSE. I suppose I'm having trouble knowing which method will be the most efficient way to solve this.
Any help would be massively appreciated!

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DrClaude
Mentor
Are the particles interacting or not? And are they bosons or fermions?

I'm not sure, this is all the question gives me. I think the question is somewhat open-ended. This is the reason I suspected there was some sort of "simple" reasoning behind the answer. Is there a method of writing out the wavelengths in a general way?

DrClaude
Mentor
If the particle are non-interacting, then $\psi$ can be written as a product of three one-particle wave functions. The latter are just the solution to the particle in a box. Care has to be taken to ensure that the final wave function has the proper (bosonic/fermionic) symmetry.

If the particles are interacting, then the problem gets much more complicated...

Yeah, I figured as much. What do you think would be the best way to proceed if I work under the assumption that the particles are interacting?

I think you should take a look at example no 5.1 page 217 of introduction to quantum mechanics 2nd ed by david.j.griffiths

That's brilliant, it's pretty much the exact question I had! Unfortunately there doesn't seem to be an answer in the book? It just sort of states the problem...

If you are asking about completeness then you can use dirichlet's theorem see page
46 of the same book