# Quantum Tunneling of a conduction electron in Copper

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1. Jan 20, 2016

### Philolaus_

1. The problem statement, all variables and given/known data
A conduction electron moves through a block of Cu until it reaches the surface. At the surface the electron feels a strong force exerted by the nonuniform charge distribution in that region. This force tends to attract the electron back into the metal which is what causes the conduction electron to remain bound to the metal. Given that the work function of the metal is 4 eV estimate the distance x that the electron can penetrate outside of the Cu block.

2. Relevant equations
Wave function = Ψ(x) = De^(-kx)
Probability of tunneling = P = |Ψ(x)^2
Wave number k = sqrt(2m(V-E))/ħ

3. The attempt at a solution
I am confident on how the penetration depth is calculated as it can be calculated using the following steps;

Compare probability at x=0 and x = Δx where Δx is the point where the probability of finding a particle is 1/e of its original value.
This gives;
1/e * |Ψ(0)|^2 = |Ψ(Δx)|^2
D^2e^(-2k*0) * 1/e = D^2e^(-2k(Δx))
1/e = e^(-2k(Δx))
-1 = -2k(Δx)
Δx = 1/2k
Δx = ħ/(2sqrt(2m(V-E)))

We know the potential is 4eV from the question, but I don't know how the energy of the electron, E, is obtained and any help on where the value for E would be appreciated..