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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

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# Derivation of reflection/transmission coefficients

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