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I am wondering about additional solutions to the particle in a box problem. In the one dimension the wave functions allowed all make sense with nodes pinned to an infinite potential barrier at either end and then divided into node increments related to the length of the box. However, in 2 or more dimensions there seam to be other solutions which I can't find a discussion or clarity on.

For example;

In a 2d equal box we could now also have wave paths at angles to the box faces. So there would be one path at 45deg to the faces with a wavelength different to those running perpendicular - but I don't see this solution discussed? There are also similar paths at 45deg where we get say a half wavelength reflected into a full wavelength (or more) - now we could assemble a wide variety of different allowable wavelengths.

Extend this principle to 3d and we would get numerous solutions.

What am I missing?