Problem Sheet: Week 9

Operators¶

1. Consider a particle whose wavefunction is the Gaussian wavepacket from Example 4.1 in the lecture notes,

   $$\Psi(x) = \left(\frac{1}{a\sqrt{2\pi}}\right)^{1/2} e^{-x^2/4a^2}e^{ip_0 x/\hbar},$$

   where $a$ and $p_0$ are arbitrary real constants.

   1. Write down the quantum mechanical prediction for the average momentum, $\langle p \rangle$, in terms of the momentum operator $\hat{P}$.
   2. Evaluate this expression for the Gaussian wavepacket $\Psi(x)$.

   In Example 5.1 we showed that the associated momentum wavefunction of the particle is

   $$\tilde{\Psi}(p) = \left(\frac{2a^2}{\pi \hbar^2}\right)^{1/4} e^{-a^2(p-p_0)^2/\hbar^2},$$

   3. Write down the definition of the average momentum $\langle p \rangle$ in terms of $P(p)$, the probability density for the particle to have momentum $p$.

   4. Evaluate this expression for the probability density $P(p) = |\tilde{\Psi}(p)|^2$ associated to the Gaussian wavepacket.

   5. Do your answers to part 2. and 4. agree with each other?

2. Consider a particle whose wavefunction is the Gaussian wavepacket $\Psi(x)$ from Problem 9.1.

   1. Calculate the action of the momentum operator followed by the position operator acting on this wavefunction, that is, calculate $\hat{X}\hat{P}\Psi(x)$.

   2. Calculate the action of the position operator followed by the momentum operator acting on this wavefunction, that is, calculate $\hat{P}\hat{X}\Psi(x)$.

   3. Write down the difference between applying the operators in the two different orders, that is, write down $\hat{X}\hat{P}\Psi(x) - \hat{P}\hat{X}\Psi(x)$.

   4. Does your result to part (c) confirm the canonical commutation relation?

The Uncertainty Principle¶

3. Consider a particle with spatial wavefunction $\Psi(x,t_0)$ and momentum wavefunction $\tilde{\Psi}(p,t_0)$ from Problem 8.3:

   $$\begin{align*} \Psi(x,t_0) &= \begin{cases} \frac{\sqrt{15}}{4}(1-x^2)& \text{if } |x| \leq 1, \\ 0 &\text{otherwise, } \end{cases}\\ \tilde{\Psi}(p,t_0) &= \sqrt{\frac{15 \hbar^3}{2\pi}}\left(\frac{\hbar \sin (p/\hbar)}{p^3} - \frac{\cos(p/\hbar)}{p^2}\right). \end{align*}$$
   1. Show that both $\Psi(x,t_0)$ and $\tilde{\Psi}(p,t_0)$ are even functions.
   2. Explain why this implies that both the average position and average momentum are zero,
      $$\begin{align*} \langle x \rangle &= 0,& \langle p \rangle &= 0. \end{align*}$$
   3. Calculate the average square position of the particle, $\langle x^2 \rangle$, and the standard deviation, $\Delta x~=~\sqrt{\langle x^2\rangle - \langle x \rangle^2}$.

   It can be shown that the average square momentum of the particle is

   $$\langle p^2 \rangle = \frac{5\hbar^2}{2}.$$

   4. Calculate the standard deviation in the momentum of the particle, $\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$.

   5. Show that the particle satisfies the Uncertainty Principle. How close is the particle to saturating the bound of the uncertainty principle?