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Section 2.6 Wave-Particle Duality

Planck and Einstein made the first big steps towards quantum mechanics by introducing photons and showing the role they play when light interacts with matter. The next breakthrough was made by de Broglie (pronounced de-BROY), who proposed what is now called wave-particle duality.

Basically, de Broglie noticed how strange the wave and photon character of light is and pondered whether this was perhaps not limited to light, but rather was a new feature of nature. Since light, which had been considered a wave, could turn out to have particle properties, perhaps things which are considered particles, like protons and electrons, could turn out to have wave properties. Perhaps everything in the microscopic world exhibits wave-particle duality.

Subsection 2.6.1 Electron Interference

What would this mean for an electron or a proton to have wave-like properties? The clear signature of waves is interference, so we could try to make electrons interfere like waves. Imagine an experiment where a beam of electrons is sent towards a double-slit apparatus, much like you did with a beam of light in lab. If the electrons are strictly particles with no wave character, we would expect to find two spots on the screen where the electrons are striking: one bright spot is the electrons that passed through the left slit and the other is the electrons that passed through the right slit. However, if the electrons are acting as waves, we should expect to see a full interference pattern on the screen, with a sequence of bright and dark spots.

Figure 2.6. Electrons are sent through a double slit apparatus and then strike a screen. The dots represent where the electrons hit the screen. As the number of electron detections grows, the interference pattern becomes clear. (Simulated data.)

These experiments have been done. We show here a simulation of the data in Figure 2.6. This figure demonstrates that once enough electrons have reached the screen, an interference pattern develops. Electrons can act as waves!

There is an extremely peculiar aspect to this experiment. First, note that the bright spots in the interference pattern are simply the regions where an electron hit more often. Equivalently, they are regions where an electron has a higher probability to hit. The electrons go through the double slit apparatus one at a time, and somehow they know where the higher and lower probability regions are. But the interference pattern is a property of both slits. For example, if the slit spacing is changed, the distance between interference bright spots is changed. This implies that somehow each single electron “experiences” both slits, since it passes through them knowing the probabilities of where to hit on the screen. This is very strange!

But very real. In fact, the wave-like property of electrons is the basis of a handy tool called the electron microscope. 1  This microscope can probe length scales significantly shorter than visible light to study, for example, cell organelles.

Subsection 2.6.2 The de Broglie Relation

de Broglie went further than just proposing this wave-particle duality; he made a specific prediction for what the wavelengths should be for things we normally think of as particles. The de Broglie relation,

\begin{equation} \lambda = \frac{h}{p}\label{eq_deBroglie_first}\tag{2.11} \end{equation}

says that the wavelength of a “particle” depends on its momentum and on Planck's constant.

Let us check that this relation works for photons: recall from the relativity unit of PHYS 211 that massless photons have an energy \(E_\text{ ph } = c|p_\text{ ph } |\text{.}\) The de Broglie relation says that the photon momentum and wavelength should be related by \(p=h/\lambda\text{,}\) so all together this gives

\begin{equation} E_\text{ ph } = c |p_\text{ ph } | = c\frac{h}{\lambda} = hf\text{.}\tag{2.12} \end{equation}

Indeed, for photons (2.11) is equivalent to Einstein's photon energy relation.

So de Broglie's proposal was that \(\lambda=h/p\) holds for all particles, not just for photons. Let's see how this works out for an electron microscope.

You wish to use an electron microscope to resolve features of a cell organelle that are on the scale of \(1\Xunits{nm}\text{.}\) To do this, you “shine” a beam of electrons on the sample. What is the minimum speed that these electrons could have?

Solution.

Recall that to resolve features on a certain scale, we need a wavelength at least that small, so we will need \(\lambda \leq 1\Xunits{nm}\text{.}\)

According to the de Broglie relation, this implies

\begin{equation} p = \frac{h}{\lambda} \geq \frac{h}{1\Xunits{nm}}\tag{2.13} \end{equation}

or

\begin{equation} p \geq \frac{6.63\times 10^{-34}\Xunits{J \cdot s}}{10^{-9}\Xunits{m}} = 6.6\times 10^{-25}\Xunits{kg \cdot m/s}\text{.}\tag{2.14} \end{equation}

Now that we have the minimum for the momentum, we can find the minimum speed simply by \(p=mv\) (we are neglecting relativistic effects):

\begin{equation} v = \frac{p}{m} \geq \frac{6.6\times 10^{-25}\Xunits{kg \cdot m/s}} {9.11\times 10^{-31}\Xunits{kg}} = 7.3\times 10^5\Xunits{m/s}\text{.}\tag{2.15} \end{equation}

So, we have seen that de Broglie argued (successfully) that particles act like waves. But waves of what? We will discuss this further in the next chapter.

Bucknell has one. Some of you may have used it.