Monthly Archives: February 2016

The Most Surprising Consequences of Quantum Mechanics

In science, today’s breakthroughs quickly become tomorrow’s monotony. Many of us use quantum mechanics everyday, but we don’t always think about its paradigm-shifting consequences and its remaining unanswered questions.

There are many online lists stating the most remarkable facts of quantum mechanics, but they often don’t adequately distinguish between the formalism and the interpretation of quantum mechanics. In my opinion, it is somewhat disingenuous to present interpretations of quantum mechanics as being part of the formalism, though this line is not always clear. The many-worlds view of quantum mechanics is a prime example that often gets media coverage. Not only is this only an interpretation of quantum mechanics, but it does not even maintain a consensus within the scientific community.

Here is a list that attempts to discuss some of what I find to be some of the most remarkable consequences of quantum mechanics. Some of these items do require some interpretation, but for these points they are at least consensus viewpoints.

1. The wavefunction is not a measurable quantity

Unlike in most other realms of physics (such as classical mechanics and general relativity), in quantum mechanics, one of the main quantities that physicists attempt to calculate cannot be directly measured. This is not because we don’t have adequate tools to do so. This is because it is not possible to do so. The wavefunction is a complex quantity, and as such, cannot be observed.

2. Quantum mechanics makes probabilistic, not deterministic, predictions

Within the realm of quantum theory, it is only possible to predict the probability of an outcome. In that sense, one does not know the trajectory of a single particle in a Young’s two-slit setup for instance, but one can predict the statistical distribution of many particles on the screen behind the slits.

3. Heisenberg uncertainty relations

This well-known theorem states that, for instance, one cannot measure the angular momentum of a particle in the x- and y- directions simultaneously without some inherent inaccuracy.

4. Identical particles, spin-statistics theorem and quantum statistics

All electrons are made the same. All photons (of same frequency) are made the same. It turns out that there are two categories of identical particles in three dimensions, bosons, which are of integer spin and fermions, which are of half-integer spin. The properties of bosons allow for superlative low-temperature phenomena like Bose-Einstein condensation. The existence of fermions and the Pauli exclusion principle, on the other hand, make sure that your hand doesn’t go through a table when you put your hand on it!

5. Non-locality, Entanglement, Bell’s Theorem

I’ve written about this on several occasions, but I will just say that quantum theory is inherently non-local. Einstein spent a pain-staking 15 years trying to make the theory of gravity local, only to see quantum mechanics pull out the rug from under is feet. See here for further details.

6. The scalar and vector potentials have measurable consequences and Berry phases

In classical electrodynamics, the quantities that have measurable experimental consequences are the electric and magnetic fields. In quantum mechanics, the Aharonov-Bohm effect demonstrates that a change in the vector potential can have experimental consequences. It is important to note that differences in potentials are gauge-invariant.

 

If you think I’ve left anything off, please let me know!

A Matter of Definitions

When one unearths a new superconductor, there exist three experimental signatures one hopes to observe to verify this discovery. These are:

  1. D.C. resistance is zero
  2. Meissner Effect (expulsion of magnetic field)
  3. Zero Peltier coefficient or thermopower

The last item is a little finical, but bear with me for a second. The Peltier coefficient effectively measures the transport of heat current that accompanies the transport of electric current. So in a superconductor, there is no heat transport (condensate carries zero entropy!), when there is electrical transport. For instance, here is a plot of the thermopower for a few iron pnictides:

thermopower

Let us ask a similar, seemingly benign, question: what are the experimental signatures one hopes to observe when one discovers a charge density wave (CDW) material?

If we are to use the superconductor as a guide, one would probably say the following:

  1. Non-linear conductivity
  2. CDW satellite reflections in a diffraction pattern
  3. An almost zero Peltier coefficient or thermopower once the CDW has been depinned

I have posted about the non-linear I-V characteristics of CDWs previously. Associated with the formation of a charge density wave is, in all known cases to my knowledge, a periodic lattice distortion. This can be observed using X-rays, neutrons or electrons. Here is an image from 1T-TaS_2 taken from here:

PLD

Now, startlingly, once the charge density wave is depinned in a large enough electric field, the thermopower decreases dramatically. This is plotted below as a function of electric field along with the differential conductivity:

thermopowerCDW

This indicates that there is very little entropy transport associated with the charge density wave condensate. Personally, I find this result to be quite stunning. I suspect that this was one of the several signatures that led John Bardeen to suggest that the origin of the charge density wave in low-dimensional materials was essentially quantum mechanical in origin.

Having outlined these three criteria, one should ask: do many of the materials we refer to as charge density waves actually exhibit these experimental signatures?

For many of the materials we refer to as charge density waves today, notably the transition metal dichalcogenides, such as 1T-TaS_2, 2H-NbSe_2, and 2H-TaSe_2, items (1) and (3) have not been observed! This is because it has not been possible to definitively depin the charge density wave. This probably has to do with the triple-q structure of the charge density wave in many of these materials, which don’t select a preferential direction.

There exist many clues that the latter materials do indeed exhibit a charge density wave transition similar to others where a depinning has been observed. It is interesting to note, though, that there are some glaring experimental absences in the transition metal dichalcogenides,  which are often considered prototypical examples of a charge density wave transition.

A Glimpse into the Renormalization Group

Leo Kadanoff passed away recently, and his ideas have had far-reaching consequences in many areas of physics. He took the fist important step of recognizing the role of “scale transformations”, which was embodied in his idea of “block spins“, at critical points. This visionary idea soon led to Wilson’s development of the renormalization group (RG).

On a personal level, I have found that understanding the concepts behind the renormalization group to be quite challenging. Many of the treatments are from a quantum field theoretical perspective, which to an experimentalist like me, present their own difficulties. I therefore found the short introductory article by Maris and Kadanoff entitled Teaching the Renormalization Group to be extremely valuable. It conveys (in 6 pages!) the main ideas behind the RG approach to phase transitions and critical phenomena.

Of course, the article only leaves the reader wanting more, as it is brief and to the point. However, I see this as a positive! It spurs curiosity. I see myself coming back to this article repeatedly even after having surmounted more difficult texts on the topic.

Just a little thought on Aharonov-Bohm Destruction of Superconductivity

Many experimenters in the past have exploited the similarities between superconductors and superfluids to come up with new ideas. One of the most important of these analogies is the Hess-Fairbank effect in a superfluid (pdf!) and the corresponding Meissner effect in a superconductor. The Hess-Fairbank effect is often taken as the pre-eminent experimental signature of a superfluid.

For those who are unfamiliar with the Hess-Fairbank effect, let me describe it briefly. It was found that if one rotates a cylindrical container of helium slowly (i.e. with angular velocity less than half the critical velocity, \omega < \omega_c/2 \equiv \hbar/2mR^2), and one cools below the transition temperature into the superfluid phase towards T \rightarrow 0, that the liquid in the container remains at rest despite the rotating walls!

We can write the Hamiltonian for the liquid in the rotating container in the rotating frame (not the lab frame!):

H'(\textbf{r}',\textbf{p}') = \frac{(\textbf{p}'-m \vec{\omega} \times\textbf{r}')^2}{2m} + V'(\textbf{r}')

Besides an unimportant centrifugal term which I’ve buried in the definition of V'(\textbf{r}'), there is a strong mathematical analogy between the Hamiltonian above and that which describes a superconductor in the presence of a vector potential (in the lab frame):

H(\textbf{r},\textbf{p}) = \frac{(\textbf{p}-e\textbf{A}(\textbf{r}))^2}{2m} + V(\textbf{r})

One can see that e \textbf{A}(\textbf{r}) corresponds to m \vec{\omega} \times\textbf{r}. Furthermore, if we assume a constant magnetic field (as in the Meissner effect), we can write for \textbf{A} = \frac{1}{2} \textbf{B} \times \textbf{r} and the correspondence becomes \vec{\omega} \leftrightarrow e\textbf{B}/2m.

Consider now what happens when one rotates the container containing the superfluid quickly (faster than the critical angular velocity). I have blogged previously about this scenario and the elegant experiment showing the proliferation of quantized vortices. In fact, if one rotates the cylinder quickly enough, the superfluidity is destroyed entirely.

Now, let us consider switching to a multiply connected geometry. It is important to note that the kinetic energy terms in the Hamiltonians remain the same if we were to use an annular container for the superfluid and a ring in the superconducting case with an Aharonov-Bohm flux through the center. For the annular container, the superfluidity would still be destroyed if the container were rotated quickly enough. This would lead one to conclude that for a superconducting ring, a strong enough Aharonov-Bohm flux would also destroy the superconductivity. There would not be an actual magnetic field applied to the superconductor (i.e. not the Meissner effect in the usual sense), but the superconductivity would be destroyed through a pseudo-Meissner effect nonetheless.

I have to admit that I have not come across an experimental paper demonstrating this effect, so please share if you know of one, as I’m sure this idea has been around for a while! It is just interesting to think that one could destroy superconductivity without actually ‘touching’ it with any kind of measurable field.

Beam Time Humor

I’m at a beam time run right now and there are some rather strange parallels with being in a Las Vegas casino. There are no windows, one often loses track of time, one gets very little sleep and the alcohol is free (though one may not want to ingest ethanol). Thankfully, my pockets aren’t empty when returning from a beam time run. Similar to other beam time runs, I find myself enjoying science jokes (don’t ask me why).

The Guardian published a selection of some favorites a few years ago. There are some real humdingers in this batch. Some highlights:

What does the B in Benoit B Mandelbrot stand for?

Benoit B Mandelbrot

What does DNA stand for?

National Dyslexia Association

And a great little limerick:

A friend who’s in liquor production,
Has a still of astounding construction,
The alcohol boils,
Through old magnet coils,
He says that it’s proof by induction.

Visualization and Analogies

Like many, my favorite subject in high school mathematics was geometry. This was probably the case because it was one of the few subjects where I was able to directly visualize everything that was going on.

I find that I am prone to thinking in pictures or “visual analogies”, because this enables me to understand and remember concepts better. Solutions to certain problems may then become obvious. I’ve illustrated this kind of thinking on a couple occasions on this blog when addressing plasmons, LO-TO splitting and the long-range Coulomb interaction and also when speaking about the “physicist’s proof”.

Let me give another example, that of measurement probability. Suppose I have a spin-1/2 particle in the following initial state:

|\psi\rangle{}=\sqrt{\frac{1}{3}}|+\rangle{} + \sqrt{\frac{2}{3}}|-\rangle{}

In this case, when measuring S_z, we find that the probability of finding the particle in the spin-up state will be P(S_z = +) = 1/3.

Let us now consider a slightly more thought-provoking problem. Imagine that we have a spin-1 particle in the following state:

|\psi\rangle{}=\sqrt{\frac{1}{3}}|1\rangle{} + \sqrt{\frac{1}{2}}|0\rangle{} + \sqrt{\frac{1}{6}}|-1\rangle{}

Suppose now I measure S_z^2 and obtain +1. This measurement eliminates the possibility of measuring the |0\rangle{} state henceforth. The question is : what is the probability of now measuring -1 if I measure S_z, i.e. P(S_z = -1 | S_z^2 = 1)? (Have a go at solving this problem before reading my solution below.)

My favorite solution to this problem involves a visual interpretation (obviously!). Imagine axes labelled by the S_z kets and the initial state by a vector in this space as so:

BeforeMeasurement

Now, the key involves thinking of the measurement of S_z^2 as a projection onto the (1,-1) plane as so:

AfterMeasurement

After this projection, though, the wavefunction is unnormalized. Therefore, one needs to normalize (or re-scale) the wavefunction again so that the probabilities still add up to one. This can done quite simply and the new wavefunction is therefore:

|\psi_{new}\rangle{}=\sqrt{\frac{2}{3}}|1\rangle{} + \sqrt{\frac{1}{3}}|-1\rangle{}

Hence the probability of measuring S_z = -1 is now increased to 1/3, whereas it was only 1/6 before measuring S_z^2. It is important to note that the ratio of the probabilities, P(S_z=1)/P(S_z=-1), and the relative phase e^{i\phi} between the |+1\rangle{} and the |-1\rangle{} kets does not change after the projection.

I find this solution to the problem particularly illuminating because it permits a visual geometric interpretation and is actually helpful in solving the problem.

Please let me know if you find this kind of visualization as helpful as I do, because I hope to write posts in the future about the Anderson pseudo-spin representation of BCS theory and about the water analogy in electronic circuits.

History and a Q&A Approach

I recently came upon the Review of Modern Physics series from the end of the 20th century where prominent figures wrote pieces about the history of subjects ranging from gravity and dark matter to biophysics and neural networks. I read Kohn’s article about the history of condensed matter physics (pdf!) from this series. Since the piece was written in 1999, it does not include a discussion of two dimensional materials, nor of topological insulators.

Nonetheless, I found the article to be quite illuminating because of the author’s historical question-and-answer perspective. Before embarking on the discussion of a topic, Kohn explained historically the importance of both the question and the answer. For example, an outstanding question at the start of the 1900s was: why do electrons in metals, which are assumed to be “free”, not contribute to the specific heat of solids at room temperature? Classically, each free electron should contribute \frac{3}{2}k_BT to the total energy by the equipartition theorem. The answer, of course, has to do with quantum statistics, which was only developed a decade later.

I have noticed that textbooks often don’t take this approach to presenting material and the important questions from a historical perspective are not explicitly stated. In my opinion, this omission leaves students often staring despairingly at the whiteboard wondering: what is the point of all of this again…? Most of us take a question-and-answer approach to our work on a day-to-day basis. It is therefore imperative that we not only relay basic material to undergraduate and graduate students, but also communicate the question-and-answer method we use to solve problems.

Can you imagine learning the photoelectric effect without understanding its  quantum mechanical consequences? Or Young’s two-slit experiment without knowing about the particle/wave debate concerning light?

Then we should not take such an approach in condensed matter physics either. For instance, Kittel’s textbook on solid state physics starts the chapter on the Debye T^3 relation with the following sentence: We discuss the heat capacity of a phonon gas and then the effects of anharmonic lattice interactions on the phonons and on the crystal. I think that this is an approach we should try to avoid. Now that I understand the historical context, I find this book to be quite valuable, but I remember struggling with it as an undergraduate.

In this sense, Kohn’s article is excellent in that it provides a historical context to some of the most important advances in condensed matter physics. It starts from classical physics, and goes through the Born-Oppenheimer theorem, the Sommerfeld model, the Bloch band paradigm, Landau Fermi liquid theory to Mott and Wigner insulators, among many other stops. I recommend it as some bedtime reading, which is exactly what it was for me.