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Amplitude coefficients are complex. The reflection and transmission coefficients must account for both amplitude change and phase change. In order to account for both of these, complex coefficients are required. These are the most general, and are needed for a complete description. In some special (and simple) cases, the phase shift is $0^\circ$ or ...

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The moller operator take the wave function and send to $t=-\infty$ with the influence of potential $V$ and send back to $t=0$ without the influence of $V$. $$\Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t }$$ If we make this with a far away wave function $\psi$, this wave function don't feel the potential $V$ at time $t=0$. But we don't ...

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The wave packet concept is also related to the inability of plane wave to describe the communication of information. The plane wave wave occupies the whole space, whereas the wave packet is localised at some position with certain distribution. This is intimately related to the group velocity and phase velocity that are also used in classical optics. To ...

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The time evolution of wave function is dictated by the Schrödinger equation, as you surely well know. Let's take the simple free particle in $\mathbb{R}^d$ (with mass $1/2$ and $\hbar=1$): $$i\partial_t \psi(t,x)=-\Delta_x \psi(t,x)\; ;$$ where $\Delta$ is the Laplacian operator (i.e. $\partial_x^2$). Mathematically, this is a linear PDE (partial ...

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The most common colour for the sky on Earth is a white sky. Clouds cover around 70% of the Earth, and the nature by which they are scattering shortwave radiation makes them appear white or grey. This is because photons coming from the Sun are likely to be scattered multiple times, and because the dependence on wavelength is not strongly increasing or ...

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This is an extremely hard question to answer definitely; its answer is heavily dependent on three things (1) the makeup of the atmosphere in question (2) the density of the atmosphere in question and (3) the surface temperature, and therefore the spectrum of output light, of the star in question. If the atmopsphere's gasses are themselves are coloured (e.g. ...

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Hmm... Mars has a red sky. But that's cause of the iron content in the soil absorbs other parts of the visible spectrum and reflects back the red part, so the sky looks reddish. I do not think a blackish sky was too far-fetched cause carbon tends to be black, smoke is blackish grey. Perceived color depends on different elements absorbing different portions ...

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We know/assume that the mass of the particle is constant. Elastic scattering means no change in kinetic energy (in the center-of-mass frame). Since the mass of the particle is constant, this means there is no change in the speed (not velocity; speed) of the particle during an elastic collision. Torque is $\vec{\tau} = \vec{r} \times \vec{F}$. For a ...

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Probably too late now, but I have to answer this. There is indeed a way to predict the direction of a scattered Lyman $\alpha$ photon. The answer depends on whether the scattering takes place in the core or the wings of the line. In the core (i.e. closer to the line center than about 3 Doppler widths), we can use the dipole approximation, so the phase ...

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Certainly late for your homework problem, but figure below shows the schematic of elastic scattering. The scattering angle is $\Theta$. The momentum transfer vector is $\Delta \bf{p}$. Since your question starts with potential, we shall obtain the force first: \begin{align} F(r) &= \frac{dV}{dr} \\ &= -\frac{\alpha}{r^2} \end{align} At any time, ...

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