Problem Sheet: Week 8

Wavepackets¶

1. In this question we will consider a wavepacket of a free particle. Assume the wavepacket has the form (4.14) from the notes, namely

   $$\Psi(x,t) = \frac{1}{\sqrt{2\pi}}\infint c(k) e^{-i\hbar k^2 t/2M} e^{ikx} dk,$$

   where $c(k)$ is given by

   $$c(k) = \begin{cases} a& \text{if } |k| \leq 1, \\ 0 &\text{otherwise. } \end{cases}$$

   and where $a$ is a positive constant.

   1. Sketch the function $c(k)$.
   2. Find $a$ such that $c(k)$ is normalised.
   3. By evaluating the above integral expression, show that the wavefunction at $t = 0$ is
      $$\Psi(x,0) = \frac{1}{\sqrt{\pi}} \frac{\sin x}{x}.$$
   4. Sketch the wavefunction $\Psi(x,0)$ and the probability density $P(x,0)$.

2. In this question we will derive the result (4.23) from Example 4.1 in the notes (Gaussian wavepackets). Assume again a wavepacket of the form (4.14) (given above), with $c(k)$ given by the Gaussian function

   $$c(k) = \left(\frac{2 a^2}{\pi}\right)^{1/4}e^{-a^2(k-k_0)^2},$$

   where $a$ and $k_0$ are arbitrary real constants.

   1. By evaluating the above integral expression for $\Psi(x,t)$ when $t = 0$, show that
   $$\Psi(x,0) = \left(\frac{1}{a\sqrt{2\pi}}\right)^{1/2} e^{-x^2/4a^2}e^{ik_0 x}.$$
   2. Write down the probability density $P(x,0)$ for this wavefunction.
   3. Make a sketch of the probability density and use it (without calculation) to find the expectation value of the position of the particle $\langle x \rangle$.
   4. Calculate the standard deviation of the probability density $P(x,0)$. That is, calculate $\Delta x = \sqrt{\langle x^2\rangle - \langle{x}\rangle^2}$, where
      $$\langle x^2 \rangle = \int_{-\infty}^{\infty} x^2 P(x,0) dx.$$

      (8.1)

Momentum¶

3. Consider a particle with the same wavefunction at time $t_0$ as Problem 7.1 of Problem Sheet: Week 7,

   $$\Psi(x,t_0) = \begin{cases} \frac{\sqrt{15}}{4}(1-x^2)& \text{if } |x| \leq 1, \\ 0 &\text{otherwise. } \end{cases}$$
   1. Show that the momentum wavefunction of the particle at time $t_0$ is
      $$\tilde{\Psi}(p,t_0) = \sqrt{\frac{15 \hbar^3}{2\pi}}\left(\frac{\hbar \sin (p/\hbar)}{p^3} - \frac{\cos(p/\hbar)}{p^2}\right).$$
      Hint

      You will need to integrate by parts twice in order to obtain this result.

   2. Sketch the wavefunction $\tilde{\Psi}(p,t_0)$ and the probability density $P(p,t_0)$.
   3. What is the probability amplitude and probability density for the particle to have momentum $p = \pi \hbar$?

4. Consider two particles, the first of which has wavefunction $\Psi(x,t_0)$, and the second of which has wavefunction $\Psi'(x,t_0)$, related to $\Psi(x,t_0)$ via

   $$\Psi'(x,t_0) = \Psi(x,t_0)e^{ik_0x},$$

   where $k_0$ is a real constant. This is the same situation considered in Problem 7.5.

   The momentum wavefunction of the first particle is

   $$\tilde{\Psi}(p,t_0) = \frac{1}{\sqrt{2 \pi \hbar}}\infint \Psi(x,t_0) e^{-ipx/\hbar} dx.$$
   1. Write down the momentum wavefunction of the second particle, i.e.~the momentum wavefunction associated to $\Psi'(x,t_0)$.
   2. Show that the momentum wavefunctions of the two particles are related via
      $$\tilde{\Psi}'(p,t_0) = \tilde{\Psi}(p-\hbar k_0,t_0).$$
      Hint

      It may be useful to introduce the new variable $p' = p-\hbar k_0$ in your answer to part 1.

   3. In a single plot, make representative sketches of $|\tilde\Psi(p,t_0)|^2$ and $|\tilde\Psi'(p,t_0)|^2$.
   4. Use your answers to part 2 and 3 to explain how the state of a particle changes when we multiply the spatial wavefunction by $e^{ik_0x}$. How does this relate to your answer to Problem 7.5.3.

5. Consider a particle with the following momentum wavefunction at time $t_0$,

   $$\tilde{\Psi}(p,t_0) = \begin{cases} \frac{1}{\sqrt{p_b - p_a}} & \text{if } p_a \leq p \leq p_b, \\ 0 &\text{otherwise. } \end{cases}$$

   where $p_a < p_b$. This wavefunction has the form of a ``box’', of width $\Delta = p_b - p_a$ and centre at $p_c = (p_a + p_b)/2$.

   1. Show that the spatial wavefunction $\Psi(x,t_0)$ of the particle at time $t_0$ is
   $$\Psi(x,t_0) = \sqrt{\frac{\hbar}{2\pi (p_b - p_a)}} \frac{(e^{ip_bx/\hbar} - e^{ip_ax/\hbar})}{ix}.$$
   2. By expressing $p_a$ and $p_b$ in terms of the centre $p_c$ and width Δ, show that the wavefunction can alternatively be written as
   $$\Psi(x,t_0) = \sqrt{\frac{\Delta}{2\pi\hbar}} e^{ip_c x/\hbar}\mathrm{sinc}(x\Delta/2\hbar),$$

   where $\mathrm{sinc}(y) = \sin(y)/y$.

   3. (Tricky) Consider now a particle with a spatial wavefunction
   $$\Psi'(x,t_0) = \sqrt{\frac{\Delta}{2\pi\hbar}}\mathrm{sinc}(x\Delta /2\hbar).$$

   Use Problem 8.4 to write down (i.e. without calculating explicitly) the momentum wavefunction $\tilde{\Psi}'(p,t_0)$ of the particle at $t_0$. What is the centre and width of this wavefunction?