Quantized Energy Levels: Understanding the Basis of Schrodinger's Equation

[Vaal]

If you solve Schrödinger's equation for a particle in a box you find the solutions to be sines and cosines. The boundary condition that the wave function must go to zero at the edges of the box then leads to the need for an integral number of wavelengths and in turn quantization of energy. I understand that some set of scalers times sine and cosine (with the correct arguments) cover all solutions to the differential equation regardless of boundary conditions. From this stand point it seems logical to me that it is not possible to have the energy be a value in which there is no solution in terms of sine and cosine. But this leads to the question of what would happen if you were to solve Schrödinger's equation using a non-allowed energy numerically?

Also, how do we know (other than empirically) that it is the energy that is quantized? Why not say quantize the mass or size of the box?

I have a pretty weak Quantum Mechanics background so this may be obvious questions, but they have been bothering me for a while. Thanks for any clarification you can provide.

 

[Avodyne]

If you try to solve the time-independent Schrödinger's equation numerically for an energy that is not an eigenvalue, you will not find a solution that also obeys the boundary conditions.