The photoelectric effect and Compton scattering show that electromagnetic waves sometimes exhibit particle-like properties. In 1923, de Broglie proposed that matter, which we normally think of as made up of particles like electrons, protons and neutrons, should have wave-like properties. In other words, everything has both wave- and particle-like properties.

Obviously, electromagnetic radiation does behave like a wave in many circumstances and matter behaves as if it were made up like particles in many circumstances, i.e., all circumstances where classical physics provides an adequate account. Therefore, whether a system exhibits wave-like or particle-like properties depends on the experiment that we are doing. This is known as wave-particle duality.

For a photon, we have $E = pc$ and the quantum postulate says that $E = h\nu$. Combining these gives $p = h\nu /c = h/\lambda$, or $\lambda = h/p$.

de Broglie proposed that the same relation should hold for matter particles, so a matter particle with momentum $p$ is associated with a wave of wavelength \[\boxed{\lambda = \frac{h}{p}.}\] This is known as the de Broglie wavelength of the particle.

de Broglie is a French name and can only be pronounced correctly by native French speakers. If you are a native English speaker then any way you choose to pronounce it is definitely wrong.

More generally, momentum $\vec{p}$ is a vector and the direction of propagation and wavelength of a wave is described by a wave vector $\vec{k}$ with magnitude given by the wave number $k = \frac{2\pi}{\lambda}$, so we want to define a relationship between the vector momentum and the wave vector. de Broglie's postulate tells us that the wave number is given by \[k = \frac{p}{\hbar},\] where \[\boxed{\hbar = \frac{h}{2\pi}.}\] The constant $\hbar$ is called the modified Planck's constant, but many people refer to both $h$ and $\hbar$ as Planck's constant, probably because $\hbar$ appears in the equations of quantum mechanics more frequently than $h$ and it would be tedious to keep saying “modified” the whole time. To avoid confusion, it is common to refer to $\hbar$ as $h$-bar. Although this is a boring name, it does lead to many extremely funny physics jokes such as “Where did the quantum physicist go for a drink after work?” and, well, I think you can guess the punchline.

Returning to more important matters, we need to upgrade the de Broglie wave number into a vector by choosing a direction. The simplest thing to do is to posit that the wave propagates in the same direction as the particle, so it has the same direction as the momentum. This gives \[\boxed{\vec{k} = \frac{\vec{p}}{\hbar}}.\] This makes sense if we imagine that a particle is somehow accompanied by a wave, because then the wave ought to be propagating in the same direction as the particle. The actual relationship between the wave and the particle, and whether either of them actually exist in the sense that we are used to from classical physics, is still controversial. We will discuss this in more detail later in the course.

If matter particles sometimes behave like waves then it ought to be possible to detect wave interference effects like diffraction and double-slit interference. At this point, you should do the in-class activity to determine the typical de Broglie wavelengths of matter particles in order to understand why it is difficult to detect these interference effects.

In 1927, Davisson and Germer scattered a monoenergetic beam of electron with energy $54\,\text{eV}$ off a slab of Nickel crystal. They observed the distribution of the intensity of the scattered electrons as they varied the scattering angle. The following figure illustrates their setup.

They found an interference

1. For this activity you will need Planck's constant $h = 6.626 \times 10^{-34}\,\text{m}^2\text{kgs}^{-1}$.

   1. Carbon has atomic mass $\approx 12 \,\text{u}$ where $1\,\text{u}= 1.661\times 10^{-27}\,\text{kg}$. Estimate the de Broglie wavelength of a Carbon 60 molecule travelling at $166\,\text{ms}^{-1}$.
   2. Estimate the de Broglie wavelength of a person of mass $70\,\text{kg}$ running at $10\,\text{ms}^{-1}$.
   3. Why is it hard to observe interference of macroscopic objects?