Motion of Particles

Abstract

In this chapter, we will explore how to describe the position and momentum of a particle in quantum mechanics, and how we set up its Schrödinger equation. To this end, we need to work out how to describe the momentum and energy for a particle. We will conclude with a description of a bizarre phenomenon called quantum tunnelling, which is now widely used in advanced microscopes, and with a qualitative overview of the quantum mechanical foundations of chemistry.

Exercises

• 1. A particle with mass m in a one-dimensional tube of length L has a Hamiltonian H that is given by

  $$ H = -\frac{\hbar ^2}{2m} \frac{d^2}{dx^2} + V(x)\, , $$

  where $V(x) = 0$ when $0\le x\le L$, and $V(x) = \infty$ otherwise (the value of $\infty$ means that the walls are impenetrable).

  • (a) Show that the wave functions

    $$ \psi _n(x) = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi x}{L} \right) $$

    are eigenfunctions of the Hamiltonian such that $H \psi _n = E_n \psi _n$, and $\psi _n = 0$ at the edges of the tube. Sketch the functions $\psi _n(x)$ for $n = 1, 2, 3, 4$.

  • (b) Show that the energy eigenvalues $E_n$ are given by

    $$ E_n = \frac{n^2 \pi ^2 \hbar ^2}{2mL^2} \, . $$

  • (c) Are the wave functions $\psi _n(x)$ eigenfunctions of the momentum operator?

  • (d) Show that functions of the form $\phi _k (x) = \exp (\pm i k x)$ are solutions to the eigenvalue equation for momentum

    $$ -i\hbar \frac{d}{dx} \phi _k (x) = p\, \phi _k (x) . $$

    Determine the relationship between k and p.

  • (e) Construct the energy eigenfunctions $\psi _n(x)$ from the momentum eigenfunctions $\phi _k(x)$. What is the physical meaning of this relationship?

• 2. Rather than working out the energy eigenstates of a particle in a one-dimensional box, we may want to study more general time evolutions of the particle. A general wave function for the particle can be written as a superposition over all the energy eigenfunctions:

  $$ \psi (x) = \sum _n c_n \psi _n (x)\, . $$

  Determine the state $\psi (x,t)$ after some time t has elapsed.

• 3. Calculate the position uncertainty $\Delta x$ for a momentum eigenstate $|{p}\rangle$, and the momentum uncertainty $\Delta p$ for a position eigenstate $|{x}\rangle$. Together with the De Broglie relation $p = h/\lambda$, this implies that wave behaviour is manifested when momentum states are prepared or measured, while particle behaviour is manifested when position states are prepared or measured. What happens when the position is measured for particles prepared in a momentum eigenstate?

• 4. By using the chain rule for differentiation, calculate an expression for the commutator $[\hat{x},\hat{p}]$ between the position and momentum operator. You will have to consider how the commutator acts on an arbitrary wave function $\psi (x)$.

• 5. Find the normalisation constant in Eq. (8.49).

• 6. Consider the wave function

  $$ \psi (x) = \frac{1}{\root 4 \of {\pi \sigma ^2}}\exp \left[ -\frac{(x-x_0)^2}{2\sigma ^2}\right] \, , $$

  with $\sigma$ a positive real number and the position x running from $-\infty$ to $+\infty$.

  • (a) What is the probability of finding a particle in the position interval $[x,x+dx)$? Sketch the probability density function $|{\psi (x)}|^2$.

  • (b) Calculate the Fourier transform $\Psi (p)$ of $\psi (x)$. What is the probability of finding a particle with momentum in the interval $[p,p+dp)$? Sketch the probability density function $|{\Psi (p)}|^2$.

  • (c) By varying $\sigma$, show how a sharper position will lead to a more uncertain momentum, and vice versa.

• 7. Derive the reflection and transmission coefficients for quantum tunnelling in equations (8.83) and (8.84). Show that these quantities behave appropriately when $V_0 \rightarrow 0$ and $V_0 \rightarrow \infty$. What happens when $a\rightarrow 0$?

• 8. The quantum state of a wave with momentum p is given in position space by

  $$ |{\psi }\rangle = \frac{1}{\sqrt{2\pi \hbar }} \int dx\; e^{ipx/\hbar } |{x}\rangle \, . $$

  express this state in momentum space.