Greetings!(adsbygoogle = window.adsbygoogle || []).push({});

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.

Thanks for your help,

DK

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Derivation of reflection/transmission coefficients

**Physics Forums | Science Articles, Homework Help, Discussion**