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RedPsi
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Homework Statement
The problem involves a particle confined to an interval 0 < x < L (in one dimension). It asks to solve for the functional form of the potential V(x) given a wave function (to get to the relevant question, I won't bother providing this). In order to solve for the potential, it was wise to use the fact that the Schrodinger equation is linear - i.e. separate the wave function into two terms and solve for V(x) for each case. In the end, the two potentials should have similar functional form.
My problem begins with the results - how to identify my constants.
Homework Equations
First potential found:
V(x) = [itex]\alpha[/itex][itex]\bar{h}[/itex] - ([itex]\bar{h}[/itex]2[itex]\pi[/itex]2/2mL2)
Second potential found:
V(x) = ([itex]\alpha[/itex] + [itex]\beta[/itex])[itex]\bar{h}[/itex] - (2[itex]\bar{h}[/itex]2[itex]\pi[/itex]2/mL2)
where alpha and beta are unknowns.
The Attempt at a Solution
From here, it appears that the second term in each expression assumes the form of energy for a typical infinite-square well problem:
[itex]\bar{h}[/itex]2[itex]\pi[/itex]2n2/mL2
where n corresponds to the quantum number indicating energy state.
My main question is, what is beta? The first term in each represents the energy eigenvalue produced by the Hamiltonian in the SEquation.
My initial thoughts:
First expression -> n = 1 state, Second expression -> n = 2 state.
1.) beta = (n-1)
OR
2.) beta = (n-1)alpha
OR
If it was real nice,
3.) beta = 0
I don't have the background knowledge on how the energy eigenfunction (first term) would look like in this perspective. Any help would be appreciated!
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