Consider a nonrelativistic electron in one dimension, with the potential(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

V(x)=\left\{

\begin{matrix}

\infty, \quad x<-a \\

-4V_0, \quad -a<x<0 \\

-V_0, \quad 0<x<a \\

\infty, \quad x>a

\end{matrix}

\right.

[/itex]

I use this as an example when I teach the Schrodinger equation in a freshman survey course. Say you're looking for a solution with an energy E=0. The momentum in the left half of the well is double what it is on the right, so the wavelength on the left is half of what it is on the right. To match the two pieces of the wavefunction without a kink, you need to have double the amplitude on the right, which corresponds to four times the probability, [itex]P_R/P_L=4[/itex].

Classically, the particle moves twice as fast on the left, so it spends half as much time there. If you peek at it at some randomly chosen moment, you find [itex]P_R/P_L=2[/itex].

Comparing these two results, you get qualitative but not quantitative agreement, and this seems to violate the correspondence principle. But in the classical limit, you can't really put the electron in a pure energy state. It's in an incoherent superposition of some large number of states with [itex]E\approx 0[/itex]. The incoherent superposition gives [itex]P_R/P_L=2[/itex], in agreement with the correspondence principle.

This would seem to be a natural model of an asymmetric diatomic molecule, but my intuition tells me that the electron would "want" to spend more time in the deeper potential, which is where it's attracted to. What's going on? The following possibilities are the ones that occur to me. (1) My intuition is just wrong. (2) The result is qualitatively different if there's a classically forbidden region separating the two sides. (3) The result only holds for [itex]E\approx 0[/itex], and for smaller energies it does what I'd have intuitively expected. (4) The result only holds because I chose [itex]V_L/V_R=4[/itex] to be a small integer, and I've been considering simple solutions where the quantum numbers are small integers.

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# Asymmetric potential well

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