(Solved): Please solve the part d and e and comment on the solutions. A steady beam of particles tra ...

Please solve the part d and e and comment on the solutions.

A steady beam of particles travels in the $$x$$-direction in a region of zero potential and is incident on a finite square well, extending from $$x=-L$$ to $$x=0$$, where the potential is $$-V_{0}$$, adjacent to a third region $$x>0$$ where the potential is $$+\infty$$, as shown in Figure 2. Each particle in the beam has mass $$m$$ and total energy $$E=V_{0} / 2$$. Figure 2 For use in Question 10 In the stationary-state approach, the beam of particles is represented by an energy eigenfunction of the form $\psi(x)=\left\{\begin{array}{ll} A \mathrm{e}^{i k x}+B \mathrm{e}^{-\mathrm{i} k x} & \text { for } x<-L \\ C \mathrm{e}^{\mathrm{i} k^{\prime} x}+D \mathrm{e}^{-\mathrm{i} k^{\prime} x} & \text { for }-L \leq x \leq 0 \\ 0 & \text { for } x>0 \end{array}\right.$ where $$A, B, C$$ and $$D$$ are complex constants. $$k=\sqrt{2 m E} / \hbar$$ and $$k^{\prime}$$ is a coefficient to be determined. (b) Use appropriate boundary conditions at $$x=0$$ to determine the relation between $$C$$ and $$D$$. (c) Use continuity boundary conditions on $$\psi(x)$$ and $$\mathrm{d} \psi / \mathrm{d} x$$ at $$x=-L$$ to obtain two equations relating $$A, B$$ and $$C$$. (d) Hence determine $$B$$ in terms of $$A, k$$ and $$k^{\prime}$$. (e) Use your expression to determine the relation between $$|A|$$ and $$|B|$$ and thus determine the reflection coefficient $$R$$.

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