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Thomson, Compton, Bloch: anniversaries

George Paget Thomson (1892-1975) died exactly 38 years ago. Arthur Holly Compton (1892-1962) was born exactly 121 years ago. And Felix Bloch (1905-1983) died exactly 30 years ago. I wrote some biographies of these men, especially Compton, in 2007.

While these September 10th men were not the ultimate fathers of the quantum theory or quantum mechanics, they surely belong to "next to the front line" of the 20th century quantum revolution. In this blog entry, I want to avoid their personal lives and focus on their key discoveries, especially in the context of the permanently persisting doubts that quantum mechanics holds everywhere in Nature.

Compton and Thomson were older. Both of them were born in 1892 and both of them are experimenters who made very important observations that made the need to replace the internal engine in the physics' heart by a new one.

Compton and the victory of photons

In 1922, Compton scattered X-rays off free electrons and observed the frequency of the resulting X-rays and the energy of the electrons. Needless to say, we know that the results obey the usual rules of the Compton scattering. All the buzzwords that carry Compton's name have something to do with this effect: Compton effect/scattering itself, Compton (wave)length (inversely proportional to the electron mass), Compton shift (change of the X-rays' wavelength).

The picture is simple. The X-ray beam is composed of photons of energy \[

E = hf = \hbar \omega = \frac{2\pi\hbar c}{\lambda} = pc

\] and the photon-electron collision obeys the momentum and energy conservation laws much like the collision of two balls. Those laws constrain the final electron's energy and the photon's wavelength. The rest is a simple calculation.

One should notice that even when Compton found his effect, there were still doubts about the reality of photons. Just to be sure, Albert Einstein received his 1921 Nobel prize in physics for his 1905 application of photons to the photoelectric effect. That derivation by Einstein made photons look "more real" than Max Planck's 1900 derivation of the black-body spectrum that used a wording meant to indicate that the derivation was just a heuristic trick.

The realization that the photon-based composition of electromagnetic waves was totally legitimate was unstoppable. Some people were ahead of others. When Compton found out that photon elastically collided like billiard balls, it was time for the remaining skeptics to evaporate. But yes, I would argue that the skeptics were already pretty painful after 1905.

Quantum field theory shows photons to be as "real" particles as electrons or any other elementary particle species. Photons are bosons while electrons are fermions; this difference makes it possible to talk about macroscopic, nearly classical electromagnetic but not Dirac fields. But otherwise the situations are completely analogous: particles are quanta of the respective fields.

Thomson and the inevitability of electrons' waves

Much like Compton, Thomson was born in 1892 but he is a September 10th guy because of the date of his death. Compton and Thomson are sort of complementary. While Compton brought us the last piece of evidence that light was composed of particles even though people would "only" view it as waves for decades or centuries, George Paget Thomson along with Clinton Davisson brought us evidence that electrons had to be associated with the wave phenomena.

In Compton's case, theorists were actually ahead of experimenters, despite the claimed "counter-intuitive" flavor of quantum mechanics. It was actually no different in the Thomson-Davisson case. They confirmed the wave properties of the electrons by their (independent) experiments in 1927, three years after Louis de Broglie conjectured that the electrons (and other particles) were associated with the de Broglie wave\[

\psi_{dB} = C\exp\zav{ \frac{i\vec p\cdot \vec x - Et}{\hbar} }.

\] De Broglie invented the right "solution", the wave function for a free particle, but all of his other comments on the issue were at least partly wrong. In 1925, Werner Heisenberg and friend established quantum mechanics; at that time, it was clear that some wave properties of the electron were needed for the atomic spectrum to be discrete. Two more years were needed for the experimental discovery of the electron diffraction.

Thomson and, independently, Davisson and Germer at the Bell Labs (Germer didn't share the 1937 Nobel prize) were firing electrons against a crystallic sample and observed some interference pattern, just like you obtain from light and gratings.

The double-slit experiment has been talked about as the ultimate "toy model for all of quantum mechanics" since the 1920s. But it actually took time for that experiment to be performed. Nevertheless, it had to work right because the electron diffraction experiments are nothing else than a "bit more convoluted" sibling of the double-slit experiment. Instead of just two slits, we have as many slits as the number of atoms or molecules in the crystal (or its surface). But the result is the same: some interference is totally crucial for any explanation of the outcome. The final probability distribution of the electron "feels" the whole structure of the crystal and its periodicities as if there were a wave involved. Indeed, a wave is involved but the right interpretation of this wave isn't a classical wave; it is a probability (amplitude) wave, the wave function.

Without Thomson and Davisson+Germer, people would really have no "sufficiently direct" detection of the particles' wave properties for many decades. Such a situation would arguably energize the "quantum skeptics" much more than in the actual history. However, I am totally sure that the development of the rest of quantum physics would proceed (be done by the true experts) even in the absence of the electron diffraction experiments.

Bloch wave and relevance of QM for solids

Felix Bloch was born in 1905, about 13 years after the experimenters above. 1905 is a sort of a bad year to be born in because he was only graduating when quantum mechanics was born so he was really too young to become its discoverer – but just by a few years. He is well-known for his contributions to NMR, the Bloch wall (a funny way in which the magnetization is changing near the boundary between two domains of a ferromagnet), and especially the Bloch wave (the same thing as the Bloch function; and its relevance for periodic potentials is the statement of the Bloch theorem).

He also invented the 1946 Bloch sphere which is a visual representation of any two-level state in quantum mechanics. You may always imagine that the states are "up" and "down" of a spin-1/2 particle such as the electron and calculate the expectation value of the intrinsic angular momentum \(\vec S\) in the wave function \((\alpha,\beta)^T\). This will unavoidably give you a unit vector in \(\RR^3\). Relatively to the axis \(\vec S\), the electron is 100% certain to have the spin "up". Bloch found this in the context of his nuclear magnetic resonance work but I think that this concept became important for the "isospin" and other two-level systems that may nicely use the mathematical isomorphism of their equations with the problem of an electron's spin.

But his 1928 Bloch wave insights – from a paper written in German, "On Quantum Mechanics of Electrons in Crystal Lattices" (a very meaningful title, by the way) – were arguably even more important. From the perspective of the quantum revolution, this insight was important because it made it clear that quantum mechanics doesn't apply just to individual atoms. It's necessary for a proper understanding of macroscopic materials, too.

Some people who feel uncomfortable about quantum mechanics often love to imagine that quantum mechanics is just something that modifies the behavior of electrons at the distance scale of 0.1 nanometers, i.e. the Bohr radius, but the electrons surely shouldn't get delocalized by millimeters or even longer distances. But they're wrong and the Bloch wave is one of the simplest proofs.

If an electron effectively only lives at points of a lattice,\[

\Gamma = \{ a_1\vec e_1+a_2\vec e_2+a_3\vec e_3;\quad \{a_1,a_2,a_3\}\subset \ZZ \},

\] the quantum mechanical problem has a symmetry group of translations that is isomorphic to \(\ZZ^3\) – triplets of integers. All the elements of the group commute with the Hamiltonian. The translations by generic vectors don't commute with the Hamiltonian so the strict de Broglie plane waves can't be the energy eigenstates for a generic \(\vec k\). But the translations by vectors from the lattice \(\Gamma\) do commute with the Hamiltonian.

This is enough to prove that the wave function (only defined at the points of \(\Gamma\)) has to be proportional to the Bloch wave \(\exp(i\vec k\cdot \vec r)\). However, one must notice that \(\vec k\) is a continuous vector here. On the other hand, shifts of \(\vec k\) by elements of the "dual" (by \(2\pi\) multiplied) lattice \(2\pi \Gamma^*\) don't change the value of the wave function for any points \(\vec r\in \Gamma\). So the wave vector \(\vec k\) is only defined as an element of the Brillouin zone of the reciprocal lattice. Léon Brillouin clarified many confusions about the periodicity conditions in his 1930 article written in French.

I can't overstate the importance of the Bloch waves and Brillouin zones in modern condensed matter physics. All the realistic pictures of the Fermi surfaces etc. depend on these concepts (and they also depend on the Fermi-Dirac statistics). The electrons in crystals and metals literally are created in states that may be approximated by the simple Bloch waves, waves that are completely delocalized. And the word "approximated" doesn't mean that the more accurate answer is "more localized" than the Bloch wave. It's not. The more accurate answer is more accurate because it acknowledges that the electrons aren't located just at the strict points of the lattice \(\Gamma\) but they also have nonzero probability amplitude to be located away from the points in \(\Gamma\) (but usually close to them because the potential energy is too high elsewhere).

So all these three September 10th men were important for the penetration of quantum mechanics into important realms of physics.

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