Problem Sheet: Week 12

Scattering¶

1. In this problem we will find the energy eigenstates of the potential step that correspond to left-moving particles. The potential energy of the particle is given by

   $$V(x) = \begin{cases} 0 & \text{if } x < 0, \\ V_0 & \text{if } x \geq 0. \end{cases}$$

   We will call the region to the left of the step, such that $x < 0$, Region I, and the region to the right, such that $x \geq 0$, Region II.

   1. Solve the time-independent Schrödinger equation separately in Regions I and II, to show that the general solution in each region is, respectively,

      $$\begin{align*} u_\rI(x) &= A e^{ikx} + Be^{-ikx}, & u_\rII(x) &= C e^{ik'x} + De^{-ik'x}, \end{align*}$$

      where $A$, $B$, $C$ and $D$ are constants, and give appropriate expressions for $k$ and $k'$.

   2. By considering what it means to be a left-moving particle, show that we must set $A = 0$.

   3. Use the continuity conditions for the wavefunction and its first spatial derivative at $x = 0$ in order to solve for $B$ and $C$ in terms of $D$.

   4. Sketch the real and imaginary parts of the wavefunction, assuming that $D$ is real and positive.

   5. Write down an appropriate definition for the reflection coefficient $R$ for a left-moving particle. Show that the reflection coefficient is given by

      $$R = \frac{(k'-k)^2}{(k'+k)^2}.$$

      How does this compare to the expression obtained in the lecture notes for a right-moving particle?

2. In this problem we will consider the same potential step from Problem 12.1, and study what happens when the particle has energy $E$ which is less than the height of the potential step, $E < V_0$.

   1. You have already solved the time-independent Schrödinger equation in Region I. Show that the general solution in Region II is now given by

      $$u_\rII(x) = Ce^{\zeta x} + De^{-\zeta x},$$

      and given an appropriate expression for ζ.

   2. By considering the behaviour of the wavefunction at $x = \infty$, argue that we have to set $C = 0$ for physically permissible solutions.

   3. Use the continuity conditions for the wavefunction and its first spatial derivative at $x = 0$ in order to show that

      $$\begin{align*} B &= \frac{ik + \zeta}{ik - \zeta}A,& D &= \frac{2ik}{ik-\zeta}A, \end{align*}$$

   4. Sketch the real and imaginary parts of the wavefunction, assuming that $A$ is real and positive.

   5. Write down an appropriate definition for the reflection coefficient $R$. Show that the reflection coefficient is given by

      $$R = 1.$$

      Does this make sense physically?

3. In this question we will find the energy eigenstates for the finite potential well that correspond to a right-moving particle. The potential energy of the particle is

   $$V(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq a, \\ V_0 & \text{if } x < 0 \text{ or } x > a. \end{cases}$$

   We are interested in solving the time-independent Schrödinger equation when $E > V_0$. We will call the region to the left of the well, such that $x < 0$, Region I, the region inside the well, such that $0 \leq x \leq a$, Region II, and the region to the right of the well, such that $x \geq a$, Region III.

   1. Solve the time-independent Schrödinger equation separately in the three regions, to show that the solution in each region is, respectively

      $$\begin{align*} u_\rI(x) &= A e^{ik'x} + Be^{-ik'x}, \\ u_\rII(x) &= C e^{ikx} + De^{-ikx},\\ u_\rIII(x) &= F e^{ik'x} + Ge^{-ik'x}, \end{align*}$$

      where $A$, $B$, $C$, $D$, $F$ and $G$ are constants, and give appropriate expressions for $k$ and $k'$.

   2. By considering what it means to be a right-moving particle, show that we must set $G = 0$.

   3. Use the continuity conditions for the wavefunction and its first spatial derivative at $x = a$ in order to show that $C$ and $D$, in terms of $F$, are given by

      $$\begin{align*} C &= \frac{k+k'}{2k}e^{-i(k-k')a}F,& D &= \frac{k-k'}{2k}e^{i(k+k')a}F. \end{align*}$$

   4. Use the continuity conditions for the wavefunction and its first spatial derivative at $x = 0$ in order to show that $A$ and $B$, in terms of $C$ and $D$, are given by

      $$\begin{align*} A &= \frac{1}{2k'}\left((k+k')C - (k-k')D\right),\\ B &= \frac{1}{2k'}\left(-(k-k')C + (k+k')D\right). \end{align*}$$

   5. Using the expressions for $A$, $C$, and $D$ from parts (3) and (4), show that

      $$F = \frac{4kk'e^{-ik'a}}{(k+k')^2 e^{-ika} - (k-k')^2e^{ika}}A.$$

   6. Using the expression for $C$, $D$ and $F$ from parts (3) and (5), show that

      $$\begin{align*} C &= \frac{2k'(k+k')e^{-ika}}{(k+k')^2 e^{-ika} - (k-k')^2e^{ika}}A,\\ D &= \frac{2k'(k-k')e^{ika}}{(k+k')^2 e^{-ika} - (k-k')^2e^{ika}}A. \end{align*}$$

   7. Using the expressions for $B$, $C$ and $D$ from (4) and (6), show that

      $$B = \frac{2i(k^2-k'^2)\sin ka}{(k+k')^2 e^{-ika} - (k-k')^2e^{ika}}A.$$

   8. Finally, show that the reflection coefficient is given by

      $$R = \frac{|B|^2}{|A|^2} = \frac{(k^2-k'^2)^2 \sin^2 ka}{4k^2 k'^2 + (k^2-k'^2)^2 \sin^2 ka}.$$