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The particular problem that triggered my question is as follows:

A particle of mass m is confined within the box $0 < x < a$, $0 < y < a$ and $0 < z < c$. The potential vanishes inside the box and is infinite outside. Find the allowed energies for a stationary state wave function of the form $X(x)Y(y)Z(z)$.

Proceeding as usual, consider the particle (mass $m$, in state $\chi$) inside the box and seek values of $E$ such that $-\dfrac{\hbar^2}{2m}\nabla^2\chi = E\chi $.

My first question is this: if necessary, how does one determine whether it is the case that $E>0$ or $E<0$ given the wording of the problem (i.e. given that there's no indication whether the state is bound or not)?

If not necessary, we proceed using the usual separation techniques and end up with a system of three second-order DE's in X, Y and Z, each with their own separation constants summing to $E$.

My next question comes in two parts: 1) Does knowing the sign of $E$ allow us to determine the sign of all of the separation constants; and 2) How does one determine which sign a separation constant takes (as in, is there a physical reason why a separation corresponding to, say, the x-coordinate must have a particular sign)?

Obviously, the sign of the separation constant makes a major difference to the solution so clearly it must be necessary to determine their signs.

Thanks in advance, this has been a niggling issue for quite some time!

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First, the particle must be in a bound state since the potential energy is infinite outside the well. Also, $E<0$ is not allowed since the minimum potential energy inside the well is zero. And while $E>0$ doesn't tell you the sign of the separation constants, the boundary conditions do: the wavefunction must go to zero at all six sides. Would that work if your eigenfunctions were real exponentials?

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