You might have seen animations like this that show an electron undergoing a transition from a lower energy to a higher energy state and vice versa like so:
There is something really important about this image that one must understand clearly.
The diagram represents the transition in energy of an electron BUT this does not mean that the electron
is magically jumping from one position and respawning at another
The electron’s position is NOT doing this i
If you want to know about the probability of finding an electron around the nucleus at a certain energy level, you look at its wavefunction and not at the energy diagram.
Here is the wavefunction of a hydrogen atom and each stationary state defines a specific energy
level of the atom.
This might not sound like a big deal but one might be surprised to know that there are a lot of people who think that the electron is magically transported from energy level to another which most certainly is not true.
If a physicist knew exactly how the universe started out, then they would be able to calculate its future for all of space and time. In this universe there is only one future which is uniquely determined by the past. The physical laws of our universe just don’t allow for more than one possible future. But a UC Berkeley mathematician has found some types of black hole where where this law completely breaks down. These claims have been made before but physicists said that a catastrophic event, such as a horrible death, would prevent observers from entering a region of spacetime where their future was not uniquely determined.
Peter Hintz, from UC Berkeley, uses mathematical calculations to show that for some specific types of black hole in a universe is expanding at an accelerating rate, it is possible to survive the passage from a deterministic universe with only one possible future, into a non-deterministic black hole.
If you did manage to travel into one of these benign singularities, then your past would be completely obliterated but it would open you to an infinite number of possible futures.
Thanks for joining us this week as we explored nearly two centuries’ worth of scientific discoveries around vibration, fluid dynamics, and quantum mechanics. For those who’d like to learn more about these and related topics, we’ve compiled some helpful resources below.
.“While the founding fathers agonized over the question “particle or “wave” de Broglie in 1925 proposed the obvious answer “particle” and “wave”.. This idea seems so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was generally ignored”
– John Stewart Bell from Bell’s Theorem
Now having taken this grand tour in pilot-wave hydrodynamics, one must also be aware of the ongoing controversy that has wrapped around pilot-wave theory over the years.
De Broglie: The pioneer of Pilot wave theory
In the eyes of De Broglie, all this would be a trip down memory lane. In 1927, he proposed an alternate interpretation for quantum mechanics – The pilot wave theory by saying that all particles are accompanied by a pilot wave.
What on earth does that mean?
Here is the analogous version of it. Observe this animation carefully:
At first, you just see a wave propagating outwards like when you drop a pebble on a pond.
But when a vibrational excitation is given, that wave is split into two traveling waves moving in opposite directions.
And as you know when two waves traveling in opposite directions are set up just right, you obtain a standing wave pattern.
This is known as a pilot wave (or) wave that pilots/guides the droplet where to go.
How does it ‘pilot’ the droplet?
At each bounce, if the droplet is made to land on the ‘incline’ of a standing wave, it would propel the droplet forward at different rates based on the level of incline.
Think of a ball hitting an inclined plane for reference
If it were to land on a flat plane, of course, it would just bounce in the same place forever like so:
All this is essential because:
De Broglie said that all particles (electrons, protons, etc) like the droplet are accompanied by physical waves that act like a pilot to guide the particle along the trajectories.
And that the pilot waves spans the entire universe.
In the 1950s Bohm took this interpretation and made it even stronger. This came to be called as pilot wave theory or Bohm-de Broglie theory or just Bohmian Mechanics.
It offers determinism that Bohr’s theory doesn’t
The most satisfying thing about this theory is that it is deterministic, i.e., one can extract sufficient information to plot a particle’s path, something that is not allowed in Bohr’s interpretation of quantum mechanics.
Bohmian interpretation applied to the Double slit experiment. Notice that the path of the particles is clearly defined and none of the particle paths cross one another but yet one obtains the same interference pattern.
For the droplet these trajectories looked like this :
All weirdness that encapsulates quantum mechanics such as wave-particle duality, wave function collapse and the paradox of Schrodinger’s cat can be avoided by using Bohmian mechanics (because it is deterministic) BUT there is a catch – nonlocality.
The pilot wave idea gives up on locality: meaning that every experiment can only be understood in the context of the entire universe. The “pilot wave” brings information from all over the entire universe to influence the event.
The cost of observing
In the series, we talked about the double-slit experiment. But here’s the deal: When you observe each electron as they are passed through the slit, the interference pattern disappears.
Disappearance of the interference pattern when observed
The way one explains this through the Bohmian interpretation is that the act of observing must obviously interfere/disturb the wave field. This, as a result, destroys the interference pattern.
Why isn’t Bohmian mechanics popular?
Sadly, the reason why Bohmian mechanics is not popular is NOT that it is scientifically inaccurate. It is able to perform equally well as other interpretations out there.
This answer by Thad Roberts does a really good job of explaining why people don’t subscribe to Bohmian mechanics. The major argument is that “It hasn’t produced anything new or predicted something better than the other interpretations.” among other critical factors.
The future for pilot-wave hydrodynamics
The droplet wave experiments remain as spectacular analogs of the pilot-wave theory at the macroscale.
But thus far, there has been no seminal evidence of pilot waves at the quantum scale.
In addition, the analogs are only capable of describing the simplest of interactions, and phenomena such as quantum entanglement are still an area of active research.
How does one weave together all of these experimental revelations that we have unearthed so far? Is there a much bigger picture of how nature manifests itself that we are yet to comprehend or are we staring at the end of a barrel?
Only time will tell.
Thank you for joining us this week on this amazing journey as we explored the essence of pilot wave hydrodynamics.
If you are thirsty to know more, FYFD will be posting a list of useful resources that we compiled, do take a look at that.
Quantum tunneling is a strange subatomic behavior that was first described to explain how alpha particles escape a nucleus during radioactive decay. Classically, a particle trapped in a well can only escape if its energy is sufficiently high, but in quantum mechanics, even a particle with lower-than-necessary energy can occasionally “tunnel” out.
To test whether hydrodynamic walkers can tunnel, researchers built corrals. In the central region, the pool on which the walker moves is relatively deep. Over the walls, the pool is much shallower. In this shallow area, the wave from the droplet’s bouncing decays quickly, creating a partially reflective barrier. For most collisions, the walker reflects off the barrier. Other times, apparently at random, a collision results in the walker crossing the wall and tunneling out of its well.
Over many experiments, researchers were able to construct a probabilistic view of walker tunneling. In quantum mechanics, a particle’s likelihood of tunneling out of a well depends on the particle’s energy and the well’s thickness. The analogs for a walker are velocity and barrier thickness. The thicker the barrier, the harder it is for a walker to tunnel through. Conversely, a faster walker has a higher probability of tunneling through a barrier of a given thickness. As the authors themselves observe:
“Although our experiment is foreign to the quantum world, the similarity of the observed behaviors is intriguing.” #
As we wrap up our series tomorrow, we’ll consider some of those similarities more deeply.
All this while, our discussion was primarily for simple quantum mechanical entities such as electrons, protons and so on.
And even for these systems merely increasing the barrier width would drastically bring down the probability.
Now if we were to scale this up to a system as complex as ours with billions and billions of atoms trying to tunnel through a wall couple of centimeters thick, nature just says ‘Sorry dude, Not gonna happen’.
Okay so maybe not the best way to break out of jail if you are a human I suppose.
But if you were a bouncing droplet, there might still be some hope. Check out the latest FYFD post on Hydrodynamic Quantum tunneling.
In quantum mechanics, the single and double-slit experiments are foundational. They demonstrate that light and elementary particles like electrons have wave-like and particle-like properties, both of which are necessary to explain the behaviors observed. Similarly, a hydrodynamic walker consists of both a particle and a wave, so, perhaps unsurprisingly, researchers tested them in both single-slit and double-slit experiments.
When a walker passes through a single-slit (top row), it’s deflected in a seemingly random direction due to its waves interacting with the slit. But if you watch enough walkers traverse the slit, you can put together a statistical representation of where the walker will get deflected. Compare that with the results for a series of photons passing through a slit one-at-a-time, and you’ll see a remarkable match-up.
If you test the walker instead with two slits, the droplet can only pass through one slit, but its accompanying wave passes through both (bottom row). Let enough walkers through the system one-by-one, and they, like their photonic cousins, build up interference fringes that match the quantum experiment. Diffraction and interference are only a couple of the walkers’ tricks, however. In the next posts, we’ll take a look at another analog to quantum behavior: tunneling.
(Image and research credits: Couder et al., source, selected papers 1, 2; images courtesy of E. Fort)
How could a light source behave like that? So you call upon your friend Dr. Tonomura (actual physicist) to conduct this experiment with photons or electrons in his laboratory to see if this behavior is consistent.
He decides to conduct it with electrons and invites you to watch. And to your astonishment, as electrons start hitting the screen you get a pattern similar to the one you got at home.
Results of a double-slit-experiment performed by showing the build-up of an interference pattern of single electrons.
Numbers of electrons are a) 11, b) 200, c) 6000, d) 40000, and e) 140000.
The pattern( known as an interference pattern ) is mysterious but similar to ones you’d seen before.
The other day when you were at the Arboretum you noticed that ripples caused by rocks thrown in the pond behave in the same way and produced the same pattern.
So what is going on?
This double slit experiment supports the idea that light is a wave since in the classical sense that you would never see such a behavior from a particle.
But then you also have experiments like the photoelectric effect which is predicated on the particle view.
So are electrons and photons behaving like a wave or a particle? Well… it’s both!
Albert Einstein wrote:
It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.
The interference pattern that we saw earlier was first observed by Thomas Young in the early 1800s. When physicists continued to study the results of the double slit, its variants and other experiments, it lead them to a bizarre new world underlying everyday reality – The quantum world. (A story for another day)
If you place a small droplet atop a vibrating pool, it will happily bounce like a kid on a trampoline. On the surface, this seems quite counterintuitive: why doesn’t the droplet coalesce with the pool? The answer: there’s a thin layer of air trapped between the droplet and the pool. If that air were squeezed out, the droplet would coalesce. But it takes a finite amount of time to drain that air layer away, even with the weight of the droplet bearing down on it. Before that drainage can happen, the vibration of the pool sends the droplet aloft again, refreshing the air layer beneath it. The droplet falls, gets caught on its air cushion, and then sent bouncing again before the air can squeeze out. If nothing disturbs the droplet, it can bounce almost indefinitely.
Droplets don’t always bounce in place, though. When forced with the right frequency and acceleration, a bouncing droplet can transition to walking. In this state, the droplet falls and strikes the pool such that it interacts with the ripple from its previous bounce. That sends the droplet aloft again but with a horizontal velocity component in addition to its vertical one. In this state, the droplet can wander about its container in a way that depends on its history or “memory” in the form of waves from its previous bounces. And this is where things start to get a bit weird – as in quantum weirdness – because now our walker consists of both a particle (droplet) and wave (ripples). The similarities between quantum behaviors and the walking droplets, the collective behavior of which is commonly referred to as “pilot-wave hydrodynamics,” are rather remarkable. In the next couple posts, we’ll take a look at some important quantum mechanical experiments and their hydrodynamic counterparts.
Next week, FYP! in collaboration with FYFD is bringing you an exclusive Tumblr series on Pilot wave hydrodynamics. There will be a new post on FYP! and FYFD all through next week (Jan 8 – 12) exploring pilot wave hydrodynamics.
This has been the topic of spectacular experimental investigations and revelations (and controversies too) in Fluid Dynamics & Quantum Mechanics in recent times.
On Monday, we begin this journey in the labs of Michael Faraday and Chladni; And then embark on an exciting adventure through decades of research to arrive at where we are today.