- #1

EE18

- 112

- 13

- Homework Statement
- Ballentine Problem 4.3 (which I am self-studying) gives is as follows:

The simplest model for the potential experienced by an electron at the surface of a metal is a step: ##W(z) = —V_0 for z < 0 ## (inside the metal) and ##W(z) =0 for z > 0## (outside the metal). For an electron that approaches the surface from the interior, with momentum ##\hbar k## in the positive ##x## direction, calculate the probability that it will escape.

- Relevant Equations
- $$-\frac{h^2}{2M}\frac{d^2\psi}{dx^2} + W\psi = E\psi \implies -\frac{h^2}{2M}\frac{d^2\psi}{dx^2} = (E-W)\psi$$

I am struggling with how to go about this; in particular, I'm not sure I understand what state is being alluded to when Ballentine says "For an electron that approaches the surface from the interior, with momentum ##\hbar k## in the positive ##x## direction, calculate the probability that it will escape." Presumably I am supposed to find some eigenstate of ##H## here, but am I to take a state with ##E>|V_0|## or ##E<|V_0|##? I would imagine we're interested in a bound state (so ##-V_0<E<0##)?