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## Homework Statement

Three particles, each of mass m, reside in a 1D "box" of length L. Find a complete set of wavefunctions,[itex]\phi_{k}(x)[/itex] for the system. In this sense, "complete" means that

*any*state of the system,[itex]\psi(x)[/itex] can be written as a superposition of the wavefunctions, [itex]\phi_{k}(x)[/itex].

## 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!