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 Originally Posted by a500lbgorilla
Share them with me.
When a particle encounters a boundary, it has a probability of "quantum tunneling" through said barrier. The solution to the wave function describing the particle experiences exponential decrease in the "forbidden" region which is the barrier.
The solution for that region is of the form
A*exp(x) + B*exp(-x)
but we can immediately rule out one of the terms, because we are going to integrate the square of the solution of the Schroedinger (to tease out ANY measurable quantity from it) from -inf to inf, i.e. over all space. In math speak, "We require all physical solutions to the Schroedinger equation to have finite L^2 norm."
So if we decide the particle is moving in the positive x-direction when it encounters this boundary, then we can rule out that part of the solution with exp(x), since that "blows up" as x goes to infinity.
So right there, we assumed that x can go to infinity, and we ruled out an entire class of mathematically viable solutions.
Then again, when we integrate the square of exp(-x) from {ongbonga} to infinity, we solve the definite integral setting exp(-inf) = 0, and we get our solution.
The solution we find uses this assumption of infinity twice and yields results correct to absurdly high degrees.
This is for the most simple cases of the SE. In general, the solutions exist in infinite-dimensional space. For instance... What is the minimum energy the electron can have in a Hydrogen atom? -13.6 eV. What's the max? Well... if it's 0, then the electron is no longer bound to the proton and it's no longer rightly an atom. There is no theoretical "highest energy level" or "biggest shell" the electron can occupy. So the solutions to the state of an electron in a Hydrogen atom exist in infinite dimensions.
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