The wave nature of particles

In 1924 de Broglie first hypothesized that associated with every particle there is a wave. In particular, we can associate a wave

$$\psi(\vec r, t) = \tilde A e^{-i ( \omega t- \vec k .\vec r )}$$ (17.11)

With a particle which has energy $E=\hbar \omega$ and momentum $\vec p = \hbar \vec k$. The corresponding wavelength $\lambda=h/p$ is referred to as the de Broglie wavelength. It should be noted that at any instant of time $t$ a particle has an unique well defined position $\vec r$ and momentum $\vec p$ . Unlike the particle, at any time $t$ the wave $\psi(\vec r, t)$ is defined all over space. This is the crucial difference between a particle and a wave. While the wave incorporates the particle's momentum, it does not contain any information about the particle's position. This is an issue which we shall return to when we discuss how to interpret the wave associated with a particle.

What is the dispersion relation of the de Broglie wave? A particle's energy and momentum are related as

$$E=\frac{p^2}{2 m}$$ (17.12)

which gives the dispersion relation

$$\omega=\frac{\hbar k^2}{2 m} \,.$$ (17.13)

Note that the relativistic relation $E=\sqrt{p^2 c^2 + m^2 c^4}$ should be used at velocities comparable to $c$.
Problem: An electron is accelerated by a voltage $V=100 \, {\rm V}$ inside an electron gun. [a.] What is the de Broglie wavelength of the electron when it emerges from the gun? [b.] When do the relativistic effects become important? (Ans: a. $1.25 \, \AA$, [b] $\sim 10^4 \, {\rm V}$)

Figure 17.3:

The wave nature of particles was verified by Davison and Germer in 1927 who demonstrated electron diffraction from a large metal crystal. A pattern of maxima (Figure 17.3) is observed when a beam of electrons is scattered from a crystal. This is very similar to the diffraction pattern observed when X-ray are scattered from a crystal. This clearly demonstrates that particles like electrons also exhibit wave properties in some circumstances.

Problems

1. Through how much voltage difference should an electron be accelerated so that it has a wavelength of $0.5 \, A^\circ$?

2. In a Compton effect experiment a photon which is scattered at $180^{\circ}$ to the incident direction has half the energy of the incident photon.

   a.

   What is the wavelength of the incident photon?

   b.

   What is the energy of the scattered photon?

   c.

   Determine the total relativistic energy of the scattered electron.

   d.

   What is the momentum of the scattered electron?

3. For what momentum is the de Broglie wavelength of an electron equal to its Compton wavelength.