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There are two heavy particles (of mass $M$) and a light one (of mass $m<<M$). The light particles interact with heavy particle with an attracting dirac delta potential V=$-\delta(q-Q_1)-\delta(q-Q_2)$, where $q$ is the position of the light particle and $Q_1$ and $Q_2$ are the positions of the heavy particles.

It's claimed that because the heavy particles exchange light particles, a potential between the heavy particles arises. Why does this occur? How do I derive this?

I'm told to find it for short and large distances. It's all one-dimensional.

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Is it a Classical or Quantum Mechanical problem? –  Vladimir Kalitvianski Dec 13 '12 at 12:59
Quantum mechanical –  Ashley_E Dec 13 '12 at 13:09
Hi Ashley. If you could edit your question to include some details about the particular concept you are stuck on, that would be good. Homework questions are usually frowned upon in this forum. Please read the homework FAQ for more information on that. –  Kitchi Dec 13 '12 at 13:58
Ok, thnx Kitchi, updated post –  Ashley_E Dec 13 '12 at 14:15
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1 Answer

Have you tried to find the ground state for the light particle and its energy's dependence on the locations $Q_1,Q_2$ of the heavy particles?

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Look like lowest energy states of light particle in this case are just a wavefunction concentrated in $Q_1$ or $Q_2$, isn't it? –  Ashley_E Dec 13 '12 at 14:19
@Ashley_E: No, it cannot be due to uncertainty principle. Also, in this symmetric case the energy level must be split. –  Vladimir Kalitvianski Dec 13 '12 at 17:02
Dear Ashley, a hint: the right wave function of the bound state is a combination of increasing and decreasing exponentials in each interval. It's continuous but the derivative has a jump. –  Luboš Motl Dec 14 '12 at 13:00
So, it should be exponentially decreasing on infinities and a constant between $Q_1$ and $Q_2$? I should find related energy, and call it's value a potential of heavy particles configuration? Thanks –  Ashley_E Dec 15 '12 at 14:02
@Ashley_E: The wavefunction will actually look like two peaks, with exponential decay from both of them. –  Dan Mar 13 '13 at 16:53
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