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Matter As Waves

When I was young, I used to think that electrons are tiny balls.

I grew up, people suggested zooming into the ball, and seeing it like this:

a wave.

Some told me that it is like a cloud:

Exploring more and more was depressing, for I did not really know much but only found what I will never know.

Somebody explained me the Uncertainty Principle like this:

Consider this:

How many of this questions can you answer with satisfaction?

1. Where is the wave?
2. What is the wavelength?

We can answer question 1 but question 2 does not make sense.

Now, have a look at this:

How many of this questions can you answer with satisfaction?

1. Where is the wave?
2. What is the wavelength?

Well, you do have an wavelength over here but the wave is pretty much spread out.

So, you either have a definable wavelength or a definable position but not both.

Electrons are waves too.
Their wavelength represents momentum:

The square of the absolute value amplitude of their amplitude gives the probability of finding the electron at that position:

$\left \langle x \right \rangle = \int x \left | \psi (x,t) \right |^2 \,dx$

So, just like you cannot have the wavelength and the position of the wave on the rope defined, you cannot have the position and momentum of the electrons well defined at the same time.

In the beginning, I wanted to blame the electrons.

Newtonian mechanics was comforting; there was nothing that occurred without a reason. For example, if you push one end of the lever,the other end goes up.You throw an object and it travels in a parabolic path.The physical world is a completely deterministic place-all the future states are derived form the previous ones.

Yesterday I was clever, I wanted to change the society
Today I am wise, I shall just change myself

After all, blaming the electrons wouldn’t be such a good idea. I made up the ideas of momentum, I thought there is something called position. Did I take the consent of these tiny guys? Nope!

To have a position, you must be somewhere. But then, matter is not anywhere, its everywhere.

They are just like those vibrating ropes except that the rope is pretty much all of the space-time:

The particles have a property ( I mean, we think they have) called it’s wave function that is defined everywhere. Schrodinger showed that they follow this equation:

We are concerned with the square of the amplitude of the wave function. It gives us the probability of finding the particle at that point. Though the wave function is time dependent, it turns out that the time dependency cancels out when we are concerned with the probability density alone.

Here is an example of a probability density plot of an electron bounded to some atom:

(Maybe time is just another parameter which we just think exists?)

God cannot play dice.
-Albert Einstein

It is amazing that something which came up just from studying gambling has given up such a broad look about the universe.

Here I will quote the author of Math With Bad Drawings to illustrate an important idea about probability:

It was a half-moon that night. The student and the teacher could see a shadowy, white-chested figure lumbering down the mountain path.
“Is that a bear?” the student gasped.

The teacher nodded calmly. “It may be. Or, it may be one of the children from the village, disguised as a bear, hoping to scare his friends.”

“Well, which is it?” the student hissed. “A deadly bear, or an innocent child?”
“Let us each determine the probability that the figure is a bear,” the teacher said.

“Then we shall share our answers with one another.”

After a pause, the student whispered her answer. “20%. It could be a bear. But it looks too short, and I think it’s wearing a backpack.”

“Very good,” the teacher said. “I say 40%. It moves slowly for a bear, but it seems to me the right size.”

“So I’m wrong,” the student said. “It’s 40%.”

“No,” the teacher replied. “You are perfectly right. For me, it is 40%, and for you, 20%.”

“But you’re the teacher. You know more.”

“And your eyes are sharper than mine. Our perspectives are different, but neither is truer. I am right, and so are you.”

“So is it a bear,” the student said, with straining patience, “or not?”
The teacher closed her eyes. “What you seek is certainty. But a probability is only a perspective. Tell me, does that creature know whether it’s a bear or not?”

“Of course.”

“So for the creature itself, the probability must be 0% or 100%. It knows with certainty. You and I have our own perspectives, and thus our own probabilities.” The teacher paused. “Tell me, if there were a full moon tonight, what would we see?”

“It’d be bright,” the student said. “We could tell at a glance if that shadow is a bear.”

“And if it were a new moon, what would we see?”

“Nothing. Darkness. There would be no shadow at all.” The student paused. “We wouldn’t see the creature approaching, so we wouldn’t even be having this conversation.”

“Precisely. When the moon is full and bright, we know all. There is no need for probability. And when the moon is new and dark, we know nothing, not even enough to ask a question. In either case – total knowledge, or total ignorance – probability is useless.

“Probability is for the nights like these,” she continued. “It is for the nights of half-light. It is for the nights when we can make out a form, but cannot tell its precise shape. It is for nights when light and shadow mingle, when knowledge and ignorance share our thoughts. It is an expression of our uncertainty – no more, no less.”

“So you’re saying,” the student said, “a probability depends on what we know, and what we don’t know. And because you and I know different things, our probabilities are different.”

The teacher smiled. Looking back out the window, the student found that the figure—bear, child, whatever it was—had vanished.

(If you really liked the story, you should read all seven of them, here)

Moral: Probability is a measure of uncertainty.

The point of the story is that probability is with respect to the observer.

The electron waves are good enough only to the one who has not yet observed them. What happens to the waves when we observe them? They collapse:

An interesting question here is “What is Observation?” or more importantly “Who can observe?

The answer is not yet well known.

To be honest, existence is the observation, not the electron.

To some, the wave collapse theory suggests that there is a universal/collective consciousness that is observing everything that exists to make them exist.

Multiplicity is only apparent, in truth there is only one mind.
-Erwin Schrodinger

You take a system of two entangled quantum particles and measure their spin as whole. Let’s say you found it to be 0. Now, you separate the particles with an arbitrarily large distance. Next, you measure one of the particle’s spin separately. Let’s say you find it is +1/2. At the same instance, you get to know that the other particle’s spin is -1/2.

The particles had not decided whether it is in a +1/2 or a -1/2 spin before it was observed, it was a superposition of both - at least that is what we just said about the matter waves. The first particle attains a certainty of its spin only when we observe it.

The paradox here is that the information that the 1st particle has attained a certain spin of +1/2 is transmitted to the other particle instantaneously, no matter where it is. Wouldn’t that imply faster than light travel of information?

As an analogy, consider that you have two envelopes that contain money. You have been told that one of them contains a 5 Rupee Note and the other contains a 10 Rupee Note. If you open one envelope and it contains a 5 Rupee Note, then you know for sure that the other envelope contains the 10 Rupee Note. It is paradoxical to note that the wave collapse in the first envelope induces the wave collapse in the second envelope immediately.

Uncertainty in quantum mechanics does not only imply a lack of our knowledge, but also a lack of a fundamental reality.

So far, we had been discussing about the wave collapse. Let us consider what the system was before we really measure it.

Let’s say three scientists, one realist, one orthodox, and one agonist do a measurement and find a particle at position P

• The realist argument: The particle was already at P. This was Einstein’s favorite viewpoint. This argument, if true, is a limitation of quantum mechanics, for it cannot give us the truth even though it exists.
• The orthodox argument: The particle wasn’t anywhere. By measuring, we forced it to take a position.
• The agnostic argument: The agnost refuses to answer. He says, “If we can only know where the particle is by measuring, then there is no such thing as a position of the particle before it was observed.” This is not so stupid after all.

Some experimental observation has ruled out the agnostic interpretations.

The most widely accepted viewpoint is the orthodox position, which is also known as the Copenhagen Interpretation.

One of the consequences of Copenhagen interpretation is the curious idea of superposition. It relies upon the fact that since a sum of solutions to Schrodinger’s Equation is also another solution, all of the solutions might represent the wave function.

We all know the famous Schrodinger’s Cat thought experiment:

I have a friend called 496 with whom I had a conversation like this.

Agnishom: Do you believe in free will?

496: What does it mean?

Agnishom: Fatalism is the belief that all our thoughts are deterministic, i.e, are exact consequences of other things.

Freewill is the belief that we have some freedom, however small, to think what we think

496: yes ofcourse…i need freedom to be alive

Agnishom: great! Fatalism is like Newtonian mechanics.

496: hmm

Agnishom: The electrons have got their free will too, maybe

The conversation soon turned into Periodic Table but that is not important in the light of this topic.

Then I met a teen theoretical physicist, Lau Chooipian in the Math Is Fun Forum who gave me his Chooipian Theory of Free Will. He believes that the randomness in the particles arise from their conscious free will.

You might consider reading his article on Quantum Consciousness on his blog.

After all, I still like the ball idea. I will end with a picture, which I like to think is of an electron.