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Consider a non-relativistic electron moving above a large, flat grounded conductor while it is attracted by its image charge, but cannot penetrate the conductor's surface.

  • What is the Hamiltonian of the electorn and the BC its wavefunction must satifsfy?

  • What is its ground state energy and its average distance above the conductor?

Classically I can write $H = T + V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right) + \frac{e^2}{4 \pi \epsilon_0} \frac{1}{2\left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2\right)^\frac{1}{2} }$

Now I would guess that the wavefunction would have to obey $\psi(x,y,z\leq 0) = 0$ and $\psi( |\vec{r}| \rightarrow \infty) \rightarrow 0$

Writing down the time indep. Schrodinger equation, however (with the QM Hamilotnian)

$E \psi(\vec{r}) = \left( \frac{- \hbar^2}{2m_e} \nabla^2 + V(\vec{r}) \right) \psi(\vec{r}) $

I am unable to find a solution since I can't use spherical symmetry, and most of the Sch. equation I have dealt with were usually for free particles (e.g. particle in box) etc. Any help would be appreicated. I'm assuming once I get the wf as a function of z I can compute its expectation value to find the average distance?

user44212
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1 Answers1

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The potential depends only on the distance to the conductor (z) not x and y. Hence it is a free particle in the x and y dimensions leaving you with the single z dimension. I believe the z equation can be related to the zero angular momentum radial functions for the hydrogen atom - recall the effective 1-D radial equation for u(r)=r R(r) is essentially a 1-d problem with and effective potential that has a centrifugal barrier term, which vanishes for zero angular momentum. Take care with the identification of r - here the distance is 2z between the charge and its image.

user3799584
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