In linear algebra, an eigenvector of a linear transformation is a non-zero vector
that only changes by a scalar factor (its eigenvalue) when that linear transformation is
applied to it.

Now for the sake of simplicity lets assume that Energy* as a linear transformation, and when it acts on some position (x1,x2) gives you the energy at that point (e0).

(x1,x2) – Eigenvector, e0 – Eigenvalue.

This e0 that you get is a physical measurable quantity and you do not want this value to be complex. Why ? Complex energies are not a thing of the real world.

And the reason why Hermitian matrices are important in Physics is because if a Matrix is hermitian, then it has real eigenvalues.

Thanks for asking!

* It need not be Energy, it could be any physically measurable quantity. We have just taken energy as an example here.

** A Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose ( A = A ^{†} )

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
position.

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.

.“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.

The phenomenon of quantum tunneling is best explained with a narrative:

You are at the bottom of a hill and need to roll a ball up the hill and down the other side.

Classically, the only option is to push the ball all the way up and roll it down. This would be a test of endurance and not to mention physically taxing.

But if you are in the quantum world you can dig a virtual hole and “probably” get to the other side of the mountain without expending as much energy as the classical ‘you’.

This behavior is called ‘tunneling’.

The reason why you would be able to pull this off in a quantum system is that there is a small probability of finding the particle in a certain location that extends to the other side of the hill.

Or if one were to put it more formally:

The wavefunction of the particle is a continous function and it cannot abruptly just collapse near the mountain/wall/barrier.

Instead it decays exponentially inside the barrier and extends onto the far side of the barrier as well.

This implies that there is a finite probability for the particle to tunnel through the barrier and get to the other side.

How big can the mountain be?

Since the wavefunction decays exponentially inside the barrier, it is no surprise that thinner the mountain, the better the chances for the particle to tunnel through.

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.

Say you are a human with a basic understanding of how the world works, i.e., you understand classical mechanics.

You decide to conduct this experiment: take a laser and shine it through a barrier with two slits. You’d expect the resulting pattern would appear something like this:

But this is not what happens! Instead, you notice this weird band pattern.

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)

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.

When one stumbles upon the words ‘Discretized solution’, one is inclined to think of Quantum Mechanics. In quantum mechanics, the following are fundamentally discrete:

Electric charge

Weak hypercharge

Colour charge

Baryon number

Lepton number

Spin

BUT not energy. One only finds discrete spectra in bound states or where there are boundary conditions.

Discrete spectra and Boundary conditions

Consider a string that is clamped at x = 0 and x= L undergoing traverse vibrations. And you would like to know the motion of the string.

Maybe you know a priori that the solutions are sinusoids but you have no information on its wave number.

So you start trying out every single possibility of the wave number.

The important thing to understand here is that If there weren’t any boundary conditions that was imposed on the string then all possible sinusoidal wave would be a solution to the problem.

But the existence of a boundary condition ruins it.

This is the case with energy as well.

If
you have an electron in a hydrogen atom, there are only specific energy
levels it can be observed to occupy when its energy is measured.

But
if the electron is unbound because its energy exceeds the ionization
energy of the atom, then it’s in a scattering state and its energy and
angular momentum have continuous spectra.

The double slit experiment is one of the most well known in modern physics. It supports wave-particle duality, which is a concept in quantum mechanics that every particle is also described partly in terms of waves.

So if a wave passes through a parallel double slit, whether it’s water, sound or light, an interference pattern will be observed. A modified version of the double slit experiment is set up to fire a single photon through a double slit such that the scientists don’t know which slit it travelled through. When the effect against the backdrop is measured, an interference pattern can be observed suggesting the photon travelled through both slits as a wave. However, as soon as they place a detector to determine which slit the photon travels through, the interference pattern disappears, and a splatter pattern can be seen against the backdrop. This shows that the photon travelled through one of the slits as a particle, and not as a waveform. So somehow by the act of observing a particle, we change it.

A modified version of the double slit experiment called the delayed choice experiment, has results that just beg more questions. At each slit they place a crystal which splits incoming photons into identical pairs. One photon from this pair will form a standard interference interference pattern and the other one will travel to a detector. Even if the photon that hits the detector is measured after the first photon hits the screen, it still changes whether or not an interference pattern was observed. So not only can we influence particles just by observing them, but those observations can alter what happened in the past.

This experiment has been repeated with molecules as large as Buckminsterfullerenes (60 carbon atoms) with the same results, and there are plans to attempt the same experiment with viruses.