# Total Transmission Across Finite Barrier Potential with E>V

1. Feb 26, 2009

### Dahaka14

1. The problem statement, all variables and given/known data
A beam of electrons of KE = 100 eV is incident from the left on a barrier
which is 200 eV high and 10 nm wide. If the momentum spread is sufficiently
narrow, then a simple plane wave is a good approximation. Recall that the mass of an
electron is mc2 = 511 keV.

.................._____
.........1.......|..2...|............3
__________|.......|____________
..............x=0....x=10 nm

(periods are there to preserve picture upon post)

If the energy of the electrons is raised to 350 eV, what thickness should the barrier be
to give transmission with no reflections?

2. Relevant equations

$$T=|t|^2= \frac{1}{1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}}$$

where $$k_1=\sqrt{2m (V_0-E)/\hbar^{2}}$$

3. The attempt at a solution

I am almost positive I just use the above equation, set $$T=1$$, then solve for $$a$$ because we know every other variable. I ended up with the general equation (no substitution for values yet)

$$1=\frac{1}{1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}}\iff 1=1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}\iff 0=\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}\iff 0=\sin^2(k_1 a)\iff n\pi=\sqrt{2m (V_0-E)/\hbar^{2}}a$$
$$\iff a=\frac{n\pi}{\sqrt{2m (V_0-E)/\hbar^{2}}}$$.

This solution makes me uneasy for some reason. Can someone point me out if I am wrong?

2. Feb 26, 2009

### lanedance

sounds reasonable, though i haven't checked your transmission equation - the transmssion condition will be when the probability waves add up constructively on transission & 100% destructively on releflection

so check this means 0 reflection..

also think about what this means for the wave number k1 (1/wavelength effectively) vs the length of the barrier...