# On Optical Tunneling and Evanescent waves

On FYP! we explored Quantum Tunneling and its hydrodynamic analogous sometime ago.

But there is an optical version of tunneling involving total internal reflection and evanescent waves that might help paint a better picture of this phenomenon.

When light encounters a different medium than the one it is already in, it bends! This phenomenon in physics is known as Refraction.

Total internal reflection
is a phenomenon which occurs when a propagating wave strikes a medium
boundary at an angle larger than a particular critical angle with
respect to the normal to the surface.

Similarly Total Internal Reflection in a prism looks like so:

Now here’s where things get interesting:

If you take a second prism and place it next to the first prism so that both glass surfaces nearly touch each other, something incredible happens.

You get a fraction of the light going through the other prism!

Source: ComputerAudioPhile

If that wasn’t clear, here is the same demonstration done with Microwaves and using wax as a prism:

## Evanescent wave

The reason why this happens is because the incident wave does not abruptly stop when it hits the end of the prism.

Some of the waves seep out with the intensity of the wave decaying exponentially outside the first prism.

But this changes when you place another prism closer to the first one.

## Does this even matter?

You might be tempted to think that this would make no difference on the macro-scale.

One possible application involving TIR and Tunneling are Optical Fibers. Loss of photons (or information) through the cladding layer of an optical fiber due to tunneling is a major concern. Therefore when they are designed the cladding layer thickness is such that the probability of tunneling to occur in such a system is extremely low.

And that’s how Tunneling, Electrodynamics and Optics are interwoven in nature.

# What is so important about Matrices being Hermitian ?

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.

* 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 )

# Note on Electron level transitions

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:

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.

The electron makes a transition between these wavefunctions by the absorption/emission of photons. *

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.

Have a good one!

– A2A

EDIT : * suggested by @imsureiforgotsomething.

# Particle and wave: A dichotomy

.“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 an event’s outcome.

## The cost of observing

In the series, we talked about the double-slit experiment. But here’s the deal: IF you observe each electron as they make their way through the slit, then the interference pattern disappers.

Disappearance of the interference pattern when electrons are 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 has posted a list of useful resources that we compiled, do take a look at that.

# Through a Quantum Tunnel

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.

## Can humans be tunneled through a barrier?

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 case you had missed out, here are the previous posts on this collaborative series on Pilot wave Hydrodynamics with FYFD 1) Introduction; 2) Chladni patterns; 3) Faraday instability; 4) Walking droplets; 5) Droplet lattices; 6) Quantum double-slit experiments, 7) Hydrodynamic Single and Double slit experiments

# A tale of Two slits

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.

PC: artemiawatson on Tumblr

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.

PC: Germain Salvato-Vallverdu

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)

Source; Tout Est Quantique

Now is it possible to replicate the same bizarre behavior that we have discussed on a macroscopic scale (say with bouncing droplets)

For this, check out the latest FYFD post on hydrodynamic double slit experiment and stay tuned for the next post in this series

In case you had missed out, here are the previous posts on this collaborative series on Pilot wave Hydrodynamics with FYFD : 1) Introduction; 2) Chladni patterns; 3) Faraday instability, 4) Bouncing droplets ,5) Complex Lattice systems

## Pilot Wave Hydrodynamics – A Tumblr series

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.

Stay Tuned!

# A note on what makes solutions discretized?

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 solution of the Schrodinger equation for the hydrogen atom

Sources and more:

Solution to the wave equation by method of separation of variables

Mathematical Methods for Physicists( Chapter – 8), George B. Arfken, Hans J. Weber, Frank E. Harris

Energy is a continuous analytic function

# Hi, about that post you made about the quantization of energy, I would like to know where you found that information, I found it very interesting but never in the two courses of quantum mechanics I took (undergraduate level) read something like that.

I first encountered it in a Carl Bender’s Mathematical physics lecture ( I’m sorry i don’t remember which one ).

Its a remarkable series on cutting edge mathematical physics. Do check it out!

EDIT : It’s on Lecture 2

# Why is energy always quantized ?

You ask a very interesting question!

The fact that energy is quantized probably feels weird to you because you are looking at it from the real domain. We must expand our horizon to the Complex plane.

## Energy is a continuous analytic function

When you move from one energy level to another, you are basically move from one Riemann surface to another and it is continuous !!!

Imagine it like those those swirling car parking places that go round and round. You get from one floor to the another rather smoothly right ?

BUT when you look at it from the outside, it feels as if you have jumped from one floor to the other, but in reality you just moved from one Reimann surface to another.

The same thing applies to energy as well.

I understand that’s a lot math. But I hope it helps 😀