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2

$e^{-i k x}$ does not blow up as $x \rightarrow -\infty.$ You're thinking in terms of real exponentials, but this is a complex exponential. That is, as long as $k$ is real we have: $\lim\limits_{x \rightarrow -\infty} e^{- k x} = \infty$ $\lim\limits_{x \rightarrow -\infty} e^{- i k x}$ does not exist (since $e^{- i k x} = \cos{kx} - i \sin{kx}$). So ...

4

This is an excellent question. The technical term for this effect is a collinear divergence. When $p_i-p_f$ tends to $0$ you get a divergence in the scattering amplitude. So why is this physically reasonable? Well remember that actual physical observables are cross-sections, not scattering amplitudes. Also recall that you cannot prepare a particle with an ...

2

As Nemis L. pointed out, the expectation value $\langle H\rangle$ is constant, because of Ehrenfest's theorem: $$\frac{d}{dt} \langle H \rangle = \frac{1}{i \hbar} \langle [ H,H ] \rangle = 0.$$ The other way of seeing this is that the state can be written as a superposition of orthogonal energy eigenstates. Obligatory image: Goldberg, Schey, and ...

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Much of scattering theory falls under the heading of Mie scattering. It examines how light scatters off uniform spheres with a given electromagnetic permittivity. In fact, it provides exact solutions to Maxwell's equations in this case. The original work was done in Mie 1908 (in German). Various further approximations can be made, leading to things like ...

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Be careful what you mean by phase here. There is an angle at which intensity is maximum $$I = \underbrace{|{\boldsymbol E}|^2}_{\mathrm{total\,\,\, measured \,\,\,field}} = I_0 \frac{\pi a^2 Q_{\mathrm{sca}} P(\theta)}{4 \pi r^2}$$ which is the mod squared of the electric field for one particle with boundary conditions within the Mie regime of the ...

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What's your problem? How do you pass from a certain vector |Q> expressed according to some base of vectors {|$B_i$>} to the same vector |Q> expressed according to another base of vectors {|$C_j$>}? Let's see it together. Take for example (1) |Q> = $∑_i$ $a_i$ |$B_i$>, where $a_i$ are the amplitudes. For passing to the new base you project |Q> on each ...

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