Problem Sheet: Week 7

Probability Density & Normalisation¶

1. Consider the following wavefunction,

   $$\Psi(x,t_0) = \begin{cases} \frac{\sqrt{15}}{4}(1-x^2)& \text{if } |x| \leq 1, \\ 0 &\text{otherwise. } \end{cases}$$
   1. Show that this is a normalised wavefunction.
   2. Sketch the wavefunction $\Psi(x,t_0)$ and the probability density $P(x,t_0)$.
   3. What is the probability amplitude at $x = \frac{1}{2}$.
   4. Where is the probability density to find the particle largest? What is the probability density there?

2. Consider the following wavefunction,

   $$\Psi(x,t_0) = \begin{cases} A\left(1+\frac{x}{2}\right)& \text{if } -2 \leq x \leq 0, \\ A\left(1-\frac{x}{2}\right)& \text{if } 0 \leq x \leq 2, \\ 0 &\text{otherwise, } \end{cases}$$

   where $A$ is a real constant.

   1. Normalise the wavefunction. That is, find the value of $A$ such that the wavefunction is normalised.
   2. Sketch the wavefunction $\Psi(x,t_0)$, and the probability density $P(x,t_0)$.
   3. What is the probability to find the particle between $x = 1$ and $x = 2$.

3. Consider the following wavefunction,

   $$\Psi(x,t_0) = \begin{cases} \frac{1}{\sqrt{a}}e^{2\pi x i /a} & \text{if } 0 \leq x \leq a, \\ 0 &\text{otherwise. } \end{cases}$$
   1. Sketch the wavefunction $\Psi(x,t_0)$, taking care to plot both the real and imaginary parts.
   2. Show that this is a normalised wavefunction.
   3. Sketch the probability density $P(x,t_0)$.
   4. What is the expectation value of the position of the particle?
   Hint

   Recall from Mathematical Physics 202 that for a probability density $P(x,t_0)$, the expectation (or expected) value is defined by

   $$\langle x \rangle = \int_{-\infty}^{\infty} x P(x,t_0) dx,$$

   (7.1)

   which is the average, or mean, position where the particle will be found.

Probability current¶

4. Consider the wavefunction

   $$\Psi(x,t_0) = \frac{1}{\pi^{1/4}} e^{-x^2/2}e^{ik_0 x},$$

   (7.2)

   where $k_0$ is an arbitrary real constant.

   1. Find the probability current $j(x,t_0)$ associated to this wavefunction.
   2. In most contexts where the continuity equation holds, the current can be viewed as being a density times a velocity (for example in fluid dynamics, electromagnetism, thermodynamics, etc).
      
      We can take this view here and write $j(x,t_0) = u(x,t_0) P(x,t_0)$, where $u(x,t_0)$ is a probability velocity. What is the probability velocity for the above wavefunction?

5. Assume that we have a wavefunction $\Psi(x,t_0)$ with associated probability current $j(x,t_0)$. Consider now a second wavefunction $\Psi'(x,t_0)$, related to $\Psi(x,t_0)$ via

   $$\Psi'(x,t_0) = \Psi(x,t_0)e^{ik_0 x},$$

   (7.3)

   where $k_0$ is an arbitrary real constant.

   1. Find the probability density $P'(x,t_0)$ associated to $\Psi'(x,t_0)$. How does it relate to the probability density $P(x,t_0)$ associated to $\Psi(x,t_0)$?
   2. Show that the probability current $j'(x,t_0)$ associated to $\Psi'(x,t_0)$ is
      $$j'(x,t_0) = j(x,t_0) + \frac{\hbar k_0}{M} P(x,t_0)$$

      (7.4)
      where $P(x,t_0) = |\Psi(x,t_0)|^2$.
   3. By considering how the probability velocity of $\Psi(x,t_0)$ and $\Psi'(x,t_0)$ are related, explain what changes about a particle when we multiply its wavefunction by $e^{ik_0x}$.

Factorising functions¶

6. Which of the following functions $f(x,y)$ factorise into $f(x,y) = g(x)h(y)$?
   1. $f(x,y) = xy^2$
   2. $f(x,y) = x^2 + 2y$
   3. $f(x,y) = xy^2 + 1$
   4. $f(x,y) = e^{x+y}$
   5. $f(x,y) = x^2\cos(y) + 2xe^{iy} + 2xe^{-iy}$
   6. $f(x,y) = \log(xy)$