Gedankenexperiments #1

Albert Einstein is well known to be one of the most creative scientists of the last couple of centuries. He produced fascinating theories that really burnished this reputation. But he also had several ideas (trying to undermine, for instance, ideas about quantum mechanics) that didn’t work – often the exact way in which they did not work led to even more insights about the theory he was trying to undermine.

Much has been written about the famous debates about the fundamental correctness of quantum mechanics and the “reality” of classical methods of describing nature. One was a debate that he carried on with Niels Bohr over several sessions, including during a famous sit-down at the Fifth Solvay Conference of 1927 (the one with the famous photo with some many scientific movers and shakers in the picture above).

The reason I wanted to write about this particular puzzle was that it is described in two different ways that fundamentally contradict each other in two popular physics books – Carlo Rovelli’s “Reality is not what it seems” and Crease/Goldhaber’s “The quantum moment”. I frankly understood neither at the first (or even second) reading. And since I am curious, here goes with my explanation.

Einstein’s thought-experiment (Gedankenexperiment in German) is very simple. He was trying to address the uncertainty relation between $\Delta t$ and $\Delta E$ . Expressed simply, this “energy-time” uncertainty relationship says that you can violate the conservation of energy (by an amount $\Delta E$ ) for a short time $\Delta t$ , as long as $\Delta E \Delta t \ge \frac{\hbar}{2}$ , where $\hbar$ is the famous Planck’s constant (divided by the number $2 \pi$ for that is what appears a lot in physics equations). Said another way, if the universe decides to create a particle-antiparticle pair out of nothing, thus violating the principle of energy-momentum conservation, it can do so – with a caveat. The larger the total energy of the particles created out of nothing, the shorter the time the particles can stick around until they recombine and sink back into nothingness. We appear to see these phenomena indirectly in Nature and the notion of “vacuum fluctuations” is well accepted in the scientific world. As an aside, what seems to be is that we calculate that we should have many more of these fluctuations that actually seem to happen – but more about that in a future post.

I don’t actually like this way of phrasing it, it seems rather mysterious. I find it easier to think in terms of Fourier components. I think of Energy as actually Frequency, using the relation $\omega =\frac{E}{\hbar}$ , that Einstein himself wrote down in his analysis of the photoelectric effect and de Broglie later used in his thesis on wave-particle duality. In that case, if I think of a function of time $f(t)$ , I could also compute its Fourier representation, which expresses the same function in terms of its frequency ( $\omega$ ) components. The above condition is then the statement of a well-known mathematical theorem (it is called the Schwarz inequality) that a function of time that is very short-lived has a large number of frequency components. Conversely, if it is very long-lived in time, it has very few frequency components.

As an example of this, suppose I played a pure note on a violin. Remember, in order for the frequency to be ${\bf EXACTLY}$ a single number, I’d have to play the note for an infinite time (a single frequency sine-wave doesn’t begin or end!). If, on the other hand, I want to describe a quick pluck of a string, I would have to include ${\bf ALL}$ the frequencies the string is capable of producing – that’s why a single plucked string immediately produces all the possible harmonics.

So, if something is short-lived in time, it has a lot of frequency components (huge “spread” in frequency space), while if it is long-lived in time, it has very few frequency components (little “spread” in frequency space).

This energy-time uncertainty relationship has been expressed in other ways. Sometimes, it is expressed as “If I want to measure an energy difference of $\Delta E$ between two states of a system, then whatever experiment I do needs to take a time period $\Delta t \ge \frac{\hbar/2}{\Delta E}$ – I cannot beat this”.

Einstein wanted to show that this version of the energy-time relation was incorrect. In particular, he wanted to show that the connection of Energy to Frequency, otherwise expressed as “Particle-Wave duality” was incorrect. Was he being inconsistent? After all, in his famous work on the photoelectric effect, he had deduced a relationship between Energy and Frequency for photons that was as written above. However, his oft-expressed thought was that this particle interpretation he had supplied for light was simply a consequence of a deficiency of theory. He believed that since he wasn’t able to construct a better theory, he had to invent a “statistical” description of light as made up of photon particles and he had to make up the above relation in such a statistical description. The short answer is that he was wrong. But the experiment he thought up is still rather interesting.

He thinks of a little box that has a shutter controlled by an on-board clock, as in the picture below.

The on-board clock is supposed to open, then close the shutter for a short prescribed time $\Delta t$ . Trapped inside the box is one photon – maybe injected much earlier by an extremely weak source of light. The energy of the photon can be arbitrarily set to $\Delta E$ . Note that we can select the numbers $\Delta t, \Delta E$ in any way we choose. In particular, we can arrange things so that $\Delta t \Delta E < \frac{\hbar}{2}$ . And to do the experiment, I simply place this contraption on a balance and wait for the weight on the balance to drop by $\frac{\Delta E}{c^2} g$ , which is the weight of the mass-equivalent of the photon. Once the photon leaks out, I’ll know immediately (though I could wait an infinitely long time to really be sure) and so I have an instance where the change $\Delta E$ was measured in time $\Delta t$ – aha!, the energy-time uncertainty relation has been violated.

Einstein supposedly presented this to Bohr one afternoon and it led to a sleepless night for the poor Dane (I should know, it led me to a few sleepless nights too and I am no Bohr, though it wasn’t at all bo(h)ring!). Clearly, if you believe quantum mechanics, something about the world should prevent you from measuring things so accurately that you know these quantities when the photon has departed its cage! And before I engage in an analysis, let me acknowledge that I benefited from a rather fruitful discussion with Scott Thomas at the physics department at Rutgers University, who might disagree with some conclusions I reached or even my approach. I take the blame for any errors.

The fiendish aspect of this experiment is that $\Delta E$ and $\Delta t$ appear to be set in stone to breach the inequality, so how could anything be amiss? I would like to take a view of this problem that a person in the 1930-50s would, so ignore any of the quantum aspects entirely. I will treat the photon as a particle, albeit one that can travel at the speed of light.

The key is that once the shutter is opened, the photon, treated as a particle, escapes. The shutter is open for a short time $\Delta t$ . If you know that a horse travelling at speed $V$ left the barn when the barn door was opened for $\Delta T$ seconds, you know that the horse could be in a region $V \Delta T$ away. Let’s ignore the possibility that the horse decided to smell the roses and was peacefully grazing outside the door, giving up the chance to escape! Similarly, the uncertainty in the position of the photon is $\delta x = c \Delta t$ where $c$ is the speed of light.

But if the photon is a particle, it is subject to the usual uncertainty principle for particles. In particular, the uncertainty in its momentum is $\delta p$ and indeed, $\delta x \delta p \ge \frac{\hbar}{2}$ . This implies that $\delta p \ge \frac{\hbar}{2 c \Delta t}$ . The energy of a photon is related to its momentum (for it is massless) by the relation $E = c p$ . It follows that the uncertainty in the photon’s energy is $\delta E \ge \frac{\hbar}{2 \Delta t}$ . Aha!, this implies that $\delta E \Delta t \ge \frac{\hbar}{2}$ .

Why is $\delta E/c^2$ also the uncertainty in the mass of the box? That’s because where could the extra energy for the photon have come from? It is not coupled to anything else! It could only have come from the box. What this implies is that the position of the photon is tied to the energy of the box, which is cryptically referred to in Carlo Rovelli’s book.

Bohr’s refutation of Einstein’s experiment, according to the Crease/Goldhaber book, boiled down to saying that $\Delta t$ would depend on the gravitational slowing of clocks due to the differing position of the box on the balance in a gravitational field. However (credit to Scott Thomas for this point), you could remove the gravitational field from this problem by simply measuring the mass of the box using a method involving a horizontal spring. The spring’s time constant would have nothing to do with the gravitational field and there would be no time “dilation” in this case. But the time constant ( $\sqrt{\frac{k}{M}}$ ) would indeed depend on the mass of the box and would serve to measure the mass. So, I am not sure why Einstein accepted Bohr’s explanation, which by the way, I wasn’t able to make sense of either.

To do this calculation in quantum mechanics, you’d have to treat the shutter as a small antenna, which emits electromagnetic fields into space. Then you would have the combined field inside and outside the box which would fall into some state which would not have definite numbers of photons inside the box. Then you would project out the outgoing states that involve a single photon and look at the spread in their energy. That would also be the spread in the energy of the box itself, since energy-momentum is conserved. But in doing so, you would use the uncertainty principle for the electromagnetic field to derive this result. Very similar to what I have done in the back-of-the-envelope argument sketched above.

References:

“The Quantum Moment” : Robert P. Crease & Alfred S. Goldhaber, W. W. Norton & Co. page 196-197.

“Reality is not what it seems”: Carlo Rovelli, Penguin Random House.

Acknowledgments: Scott Thomas @ Rutgers