2.2 The Matter Wave

2.2.1 Wavelength

In 1924, Louis de Broglie hypothesized the matter wave. The wavelength associated with a massive object is given by

$$ \bbox[silver]{\lambda=\frac{h}{p}.} $$

This relationship is true in general, even for relativistic cases. This contribution won for de Broglie the 1929 Nobel Prize.

2.2.2 Frequency

The frequency of a matter wave follow the same relationship as for electromagnetic radiation

$$ \bbox[silver]{f=\frac{E}{h}.} $$

Recall that the wave number (or wave vector, which can be understand as "spatial frequency") and angular frequency are defined as $k=2\pi/\lambda$ and $\omega=2\pi/T$ respectively, it would be convenient to define

$$ \bbox[silver]{\hbar=\frac{h}{2\pi}\approx1.055\times10^{-34}J\cdot s.} $$

Then the wave-particle properties can be linked via

$$ \bbox[silver]{p=\frac{h}{\lambda}=\hbar k,} $$
$$ \bbox[silver]{E=hf=\hbar\omega.} $$

Again, these two relations are true in general.

2.2.3 Velocity

The relation $v=f\lambda$ can be directly applied to find the speed of matter wave

$$ v_{phase}=f\lambda=\frac{E}{h}\frac{h}{p}=\frac{E}{p}. $$

However, we should notice that this is the velocity of the wave, or the phase velocity. The velocity of the particle, or the group velocity can be found using energy and momentum.

Practice Problems for 2.2

  1. (Harris) A Bragg diffraction experiment is conducted using a beam of electrons accelerated through a 1.0 kV potential difference.

    (a) If the spacing between atomic planes in the crystal is 0.1 nm. at what angles with respect to the planes will diffraction maxima be observed?

    (b) If a beam of X-rays produces diffraction maxima at the same angles as the electron beam. what is the X-ray photon energy?

  2. (Harris) Classically and nonrelativistically, we say that the energy $E$ of a massive free particle is just its kinetic energy.

    (a) With this assumption, show that the classical particle velocity $v_{particle}$ is $2E/p$.

    (b) Show that this velocity and that of the matter wave differ by a factor of 2.

    (c) In reality, a massive object also has internal energy no matter how slowly it moves, and its total energy $E$ is $\gamma mc^2$, where $\gamma\equiv1/\sqrt{1-(v_{particle}/c)^2}$. Show that $v_{particle}$ is $pc^2/E$ and that $v_{wave}$ is $c^2/v_{particle}$. Is there anything wrong with $v_{wave}$ ?