(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A point charge Q with mass M approaches a semi-infinite surface of a perfect metal and becomes trapped near the surface in a (quantum) bound state. Find the binding energy of this particle to the surface. Treat the metal classically—i.e. ignore its internal quantum levels, etc., and just consider it to be a classical perfect conductor that the point charge’s wavefunction cannot penetrate.

2. Relevant equations

The 1D Time-independent Schrodinger equation.

[tex] \frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}} +V(x) = E\psi [/tex]

3. The attempt at a solution

My problem with this question mostly stems form the nature of the potential surrounding the conductor. If we assume that the charged particle is approaching the surface due to the presence of an electric field and we treat the surface as a plane then the potential is of the form:

[tex] V(x) = QEx = \frac{Q\sigma}{\epsilon_{0}x [/tex]

Where sigma is the surface charge of the surface.

Or we can consider the particle to be influenced by an induced surface charge, which would take the form of a negative mirror charge (-Q) so the potential would be a simple Coulomb attraction.

[tex] V(x)= \frac{-Q}{4\pi\epsilon_{0}r} [/tex]

Where we would have to account for the change in separation as the charge approached the surface.

My question is a) Which of these options seems the most reasonable. My guess is the second one but I'm not sure I feel I'm missing something.

b) For the charge to become bound the potential needs to be "well-like" i.e. I need a repulsive component for distances very close to the surface of the conductor. I'm afraid I don't know what form this component could be. Since the wavefunction cannot penetrate the surface of the conductor I would think we could treat the potential as infinite for x<0 (putting the conductor at x=0). However, this would mean the well had no "bottom" and the charge would not bind.

I hope that's clear. Any help resolving my confusion would be greatly appreciated.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Potential of a charge approaching a perfect conductor.

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