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Date: 14-8-2016
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Date: 25-7-2016
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Charge and Plane
A particle moving in one dimension feels the potential
(This potential would be appropriate for an electron moving in the presence of a uniformly charged sheet where C is the transparency of the sheet.)
a) Using the WKB approximation, find the energy spectrum, En, for this one-dimensional problem for all n for C = 0.
b) Find the energy spectrum, En, for C → ∞.
c) Derive an equation that describes the energies En for even wave functions for an arbitrary value of C. What can you say about the energies En for odd wave functions?
SOLUTION
a) Since V(x) = γ|x|, we may write
(1)
In the WKB approximation
or, between turning points,
(2)
Substituting (1) into (2) and using the symmetry of the motion
about x = 0, we obtain
(3)
Thus,
(4)
(5)
b) For the potential where C → ∞,
the quantization condition gives
(6)
(7)
(8)
c) Using the boundary conditions at x = 0, we obtain
(9)
It implies that the odd states, for which ѱ(0) = 0, are not affected by Cδ(x), while even states should satisfy the condition
(10)
Since
(11)
where
(12)
this condition takes the following form:
(13)
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