ShamelessGit
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I tried to solve a time independent schroedinger equation with a finite potential well today by solving it in 3 pieces, one for in the box and 2 for the outsides. By setting the equations equal to each other where they met at the edges of the box, by setting the integral of everything squared = 1, and by setting the integral of the exponential functions to the right and left equal to each other (I figured it made sense for them to by symmetrical), I was able to write all the constants equal to 1 other constant, which I could set equal to a function involving energy. I don't like this because it looks as though the constants and energy could vary infinitely, but energy is supposed to be quantized. It seems intuitive to make it a smooth function, but I don't ever remember being taught that one is supposed to do that. I tried making it smooth and it looks as though I get convenient answers.
1. Is it appropriate to solve the Schrödinger equation in pieces like this?
2. Is it good to make it smooth?
3. An unrelated question: Is there also a corresponding time dependent equation? I'm familiar with separation of variables, and the full Schrödinger equation makes it look as if there should always be a time dependent solution, and if there's not, then the total energy E is equal to 0. When, if ever, is there a wave-function that is independent of time?
4. Is there anything else I missed?
1. Is it appropriate to solve the Schrödinger equation in pieces like this?
2. Is it good to make it smooth?
3. An unrelated question: Is there also a corresponding time dependent equation? I'm familiar with separation of variables, and the full Schrödinger equation makes it look as if there should always be a time dependent solution, and if there's not, then the total energy E is equal to 0. When, if ever, is there a wave-function that is independent of time?
4. Is there anything else I missed?