A solvable model for the finite rectangular potential well with a bump in the middle

Question

A well known example in quantum mechanics is that of a finite rectangular potential well with a rectangular bump in the middle. I guess this closely approximates the "umbrella" effect of the $NH_3$ molecule.

But this potential is not solvable analytically.

• I want to know if there is a solvable Hamiltonian known which mimics the effects of this potential - like from which one can exactly see the effect on the energy levels and the wave-functions of the width of the bump or the height of the bump or the well width on either side of the bump.

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Comments

It depends what you call solvable analytically. The potential you described yields a transcendental equation in a single variable: the energy. Of course this equation can't be solved analytically but this is quite good as far as Shrodinger equations go

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@Squark By solvable - I mean exactly solvable :) From that transcendental equation is it possible to see any of the dependencies that I mentioned? I have seen at places that the kind of potential I talked of is modelled by a $2 \times 2$ matrix such that the $11$ and the $22$ component has say $H_0$ and the both the off-diagonal elements have say $-\Delta$. This is solvable exactly and does help reproduce some of the effects like the fine double splitting of the energy (typical of the Ammonia spectrum) as $\Delta$ is moved from $0$ to non-zero.

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@Squark But this model Hamiltonian is still not sophisticated enough to capture the effect of the width and height and position of the bump on the spectrum and wave-functions. Hence I am looking for something better. Like one would want to make statements about how fast the probability density maxima oscillates from the right to the left as a function of the width, height and position of the bump - or how the time evolution is affected by what kind of linear superposition of states (say the ground and the first excited) or whatever state one starts off with.

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Firstly, it is conceivable, even likely, that by analyzing the equation it is possible to understand the qualitive dependence of the energy levels on the various parameters. Other quantites are derivable from this, for example the freq. of oscillation between the two minima is ~ h times the energy difference between the odd and even groundstates. I think that since you're interested in qualitive features only the WKB approximation might be appropriate

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@Squark May be you can elaborate more on how you think these fine details can be gotten without actually solving - as you seem to suggest.

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