A particle in a one-dimensional quantum well is governed by a variant of the time-independent Schrodinger equation, expressed in terms of wave function Ψ ( x ) .
The quantities V and E are the potential energy and total energy, respectively.
− d x 2 d 2 Ψ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x )
The potential varies as follows:
V ( x ) = ∞ x < − π V ( x ) = cos 2 x − π ≤ x ≤ π V ( x ) = ∞ x > π
The boundary conditions on Ψ ( x ) are:
Ψ ( x ) = 0 x ≤ − π Ψ ( x ) = 0 x ≥ π
Let E 1 , E 2 , and E 3 be the lowest, second lowest, and third lowest allowable values of E , subject to the constraint E > 0 . Determine the following ratio:
E 2 − E 1 E 3 − E 2
Note: Only apply the Schrodinger equation within the finite potential region
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I used the same procedure outlined by @Karan Chatrath . What we want to do is make a graph of E vs. the value of Ψ at the right end of the well, assuming that we initialize properly at the left end of the well for each value of E . The allowable values of E are those for which the graph crosses the horizontal axis. The code for making the graph is attached.
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Very nice! From the looks of the plot, the gap between successive allowable energies increases as energy increases.
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Indeed. I was thinking back to the simple potential-free textbook quantum well. And the energy there is proportional to n 2 , where n = 1 , 2 , 3 , e t c . . The gaps between energy levels increase there too, as a result of the n 2 dependence. Just for fun, I ran this same code with V = 0 and reproduced the results for the classic quantum well, showing the n 2 dependence. The ratio in that case is 4 − 1 9 − 4 = 3 5 .
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Interesting. Running the V = 0 case is a good way of cross-checking one's code cause the results are well known. I say this especially since in the last problem, my code had some errors.
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This problem was treated purely numerically. Steps performed are as follows:
A plot showing the wave functions at different allowable energy levels is as follows: