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I have learned about the infinite quantum well last week and today the finite quantum well was introduced. There is however one thing I don't completely understand, or which I'm not completely happy with.

In the case of the finite well, the shrodinger equation is solved for both inside and outside the well, yielding the usual sine/cosine inside and now a decreasing exponent outside the well, meaning the particle can leak through.

In the case of the infinite well, instead of 'trying to solve' the shrodinger equation, it is simply stated that [itex]\psi = 0[/itex] outside the well.

Is this all there is to it, or can you determine this by actually trying to solve the shrodinger equation (although I have no idea how that would work) using an infinite potential?

If you can't calculate it using the shrodinger equation, is the fact that the potential is infinite enough to make it absolutely certain that the particle cannot be there? Classically, the particle cannot be outside a finite well either (assuming it does not have enough energy) but by solving the shrodinger equation we can show that it can, so when I hear this quick 'psi = 0' argument without further elaboration, something does not 'agree' inside my head...

It seems that either people simply assume that because the potential is infinite, the particle can't be there, or that there is something else going on that was not explained to me yet (i hope the latter...)

I just feel that, in this 'all new world' of quantum physics where strange effects are so common, you shouldn't just say that the particle can't be outside the infinite well without further explaining...