# Derivation of reflection/transmission coefficients

1. Dec 28, 2013

### Delta Kilo

Greetings!

Going throught Ballentine Ch.4 and the derivation of transmission/reflection coefficients. The math seems fairly straightforward: assuming a particle in a piecewise-constant potential, the solution of the correspondent time-independent SE is a piecewise-exponential wavefunction in a form $ψ_i = A_i e^{k_ix} + B_i e^{-k_i x}$, where $k_i=\sqrt{-2m(E-V_i)}/\hbar$ is either real or imaginary depending on the sign $E-V_i$. After adding corresponding boundary conditions, one can solve the system for the allowed values of $E$ and corresponding $A_i$ and $B_i$. So far so good.

Now here is the bit I'm not comfortable with: we treat the two parts $Ae^{ikx}$ and $Be^{-ikx}$ separately, and claim that one describes the particle going left-to-right and another right-to-left. Why? As far as I can see, there is just 1 particle and it is not going anywhere because the solution is time-independent. And it is just a sheer luck that I get a solution as a sum of "left" and "right" parts. What if I get quadratic potential wells instead of boxy ones, then the solutions will be gaussian-ish (energy states of harmonic oscillator to be exact), with no obvious split into "left" and "right" parts.

I mean it all seems sort of right intuitively, but I have a feeling there is something they are not telling me. For example, problem 4.3 asks: electron with momentum $p=\hbar k$ going from left to right, impinges on a potential step of height V, what is the probability of it passing through. Ok I know (I think) how to calculate the answer they expect, which is the transnission coefficient , I just dont see why it should be the answer. They ask for probability and the only official recipe given so far in the book for calculating a probability is $Prob\{R<a\}=\left\langle \theta(R-a) \right\rangle = Tr\{\rho \theta(R-a)\}$ where $\rho$ is a state operator, $\theta$ is a unit step function and $R$ is an observable. I just don't see how to apply it to the problem at hand.