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(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

(b) Use appropriate boundary conditions at \( x=0 \) to determine the relation between \( C \) and \( D \).
(c) Use continuit

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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Schrodinger's equation for the region x
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