- #1

nonequilibrium

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I'm in my last year of bachelor of physics and following a QM class and as is standard we calculated the reflection R and transmission T coefficients for a plane wave in some potential well/barrier situation.

Our teacher said "of course, the plane wave is not a good genuine wave function, not being normalizable, but due to superposition we can make a wave packet and the R and T are also applicable to this wave packet."

I had some objections to this quote and upon inquiring more, he said that for a wave packet described as [itex]\psi = \int g(k) e^{i(kx-\omega(k) t)} \mathrm d k [/itex] the reflection coefficient (for example) is [itex]R = \int g(k) R(k) \mathrm d k [/itex] where R(k) is the earlier calculated reflection coefficient for a plane wave with parameter k.

I get the intuitive plausibility that the above formula has, but when I try to think about it more exactly and try to argue it mathematically it seems to be untrue.

My reasoning is as follows: how does "R" acquire its meaning as some number characterizing how much goes back? Due to the notion of probability current. So for the above formula to work, the probability current of all the plane waves must also adhere to the superposition principle, i.e. [itex]J = \int g(k) J(k) \mathrm d k[/itex], and this is what my professor (literally) said when I asked him. However, when I look at the definition of J, namely something proportional to [itex]J \propto \psi^* \nabla \psi - \psi \nabla \psi^* [/itex], that looks very unlinear (w.r.t. to psi), so how can we possibly rely on the superposition principle?

Thank you!