# Matter as Waves

Particle electron

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:

Wave

a wave.

Some told me that it is like a cloud:

Electron 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: The Pulse

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: The Wave

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:

deBroglie Waves

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

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

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.

Uncertainty Principle

Heisenberg at home

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:

Waves

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:

Schrodinger's 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:

animated

(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:

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:

Schrodinger's Cat

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.

Periodic Table

Then I met someone in the Math Is Fun Forum who gave me his Theory of Free Will. He believes that the randomness in the particles arise from their conscious free will.

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

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

Smiley

5 years, 1 month ago

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Why am I not able to see several parts of it- Like your friend's name written here, also when I try to look aqt people's activities, and see their statuses changed- It appears blanked out similar to what is happening here...Is it only me...or? @Calvin Lin

- 5 years, 1 month ago

YES!!! I am also facing this problem.

- 5 years, 1 month ago

Which friend? I just called her 496

Staff - 5 years, 1 month ago

same with me

- 5 years ago

Can you provide more explanations of what you mean?

• The friends name is just denoted by " > 496:", as opposed to an actual name.
• Can you send me a screen shot of what "it appears blanked out" mean?

Staff - 5 years, 1 month ago

Ok, I will . But I think there is some problem with the display here.

- 5 years, 1 month ago

Dimension of gauge group of string theory.. and also the 496 is considered to be perfect number..

- 5 years ago

What is the gauge group?

Staff - 5 years ago

Some pics are not appearing

- 4 years, 9 months ago

It's my fault. I will fix it as soon as I can.

Staff - 4 years, 9 months ago

Oh! Ok! Thanks for the reply! Nice note!

- 4 years, 9 months ago

Basically, I mean a pulse has a well defined position but not a wavelength. Whereas, the opposite is true for a wave

Staff - 4 years, 9 months ago

Yeah! I got that! Btw are you preparing for JEE?You seem to be in 11th or 12th.

- 4 years, 9 months ago

Great article thanks for sharing @Agnishom Chattopadhyay .

- 5 years, 1 month ago

Easy to read and most of the time easy to understand. Great article

- 5 years, 1 month ago

Thanks, I have included some equations for the geeks out here but you can still understand the text without them.

Staff - 5 years, 1 month ago

What an article !! I especially liked the way you explained the uncertainty principle. Keep up the good work.

- 5 years, 1 month ago

Guys, I wrote this for an assignment my Physics teacher gave me. I hope it is kind of okay

Staff - 5 years, 1 month ago

That was $O\sum$. All great things are simple! I remember that the definition of force kept on complexing from 4th to 10th class, and in 11th class, it again became the same good old one, "A push or a pull is called a force.". Remember, "A dot is a dot."

- 5 years, 1 month ago

What is O Sigma?

That definition keeps you wondering what a push or pull is

Staff - 5 years, 1 month ago

Oh! That is a short form for "awesome", I have the copyright... :D

- 5 years, 1 month ago

O-sum! LOL :D

- 5 years, 1 month ago

Yeah! BTW I'm now on.. :D

- 5 years, 1 month ago

Thanks :) Can someone out here do me a favor by re sharing this note? I mistakenly saved it in a set and I cannot get my followers to see it :(

Staff - 5 years, 1 month ago

No Worry! I loved the note, and I shall share it!

- 5 years, 1 month ago

It just became popular. Wow! Thanks for the bump :)

Staff - 5 years, 1 month ago

Tried Bashing Available

- 5 years, 1 month ago

Alright before I read this lemme state my knowledge:

Conceptual Physics (Mechanics, Thermodynamics, Properties of Matter, Electromagnetism, Light, Nuclear Physics, Relativity) (algebra math only)

Calculus AB + BC + some vector calc + learning Multivariable

- 5 years, 1 month ago

Yes.

- 5 years, 1 month ago

Ty sir :)

- 5 years, 1 month ago

Staff - 5 years, 1 month ago

Nice indeed. But I to some extent support the realist argument,for the position of an electron or any other particle is independent of who is measuring it or how it is being measured.

- 5 years, 1 month ago

Wow this is so long

- 5 years, 1 month ago

My apologies.

Staff - 5 years, 1 month ago

why?

- 5 years, 1 month ago

For making it long.

Staff - 5 years, 1 month ago

Altering the laws :P liked the wy you explained

- 5 years, 1 month ago

Amazing thanks for shari

- 5 years, 1 month ago

A beautiful article, I must say

- 5 years, 1 month ago

Great job done bro...thanks a lot....

- 5 years, 1 month ago

Awes0me report!!! Really loved it

- 5 years ago

Great bro....very well...nd thanks for the links

- 5 years ago

what an article!

- 5 years ago

It's been a long time since I've read a really good topic on science......thanks for this post.

- 5 years ago

I am myself a 16-year, so I felt the what you did....the topic of matter waves was quite revolting to me at the beginning, but after I read about de broglie's equation, i thought I should have done it before!!!

- 5 years ago

very interesting and thought provoking,continue the good work

- 5 years ago

Thanks for not over doing it like most people do.

- 5 years ago

Overdoing what?

Staff - 5 years ago

absolutely brilliant, tried to comprehend as much as possible, world has got a few brilliant inhabitants in the 6.6 bln. reading the brilliant piece certainly increase my neural net work, the piece on the probability of " to be or not to be a bear " has taught me to reduce the uncertainities as much as possible to get a p as close to 1 to be a winner on the stcok markets.

- 5 years ago

really really great article!Very insightful...........

I understood the Uncertainty Principle fully after seeing this note. Keep sending such notes on Quantum Mechanics!!!!!

- 5 years ago

Hey Agnishom, I think you should do some corrections. I did not believe that particles have their free will. I believe <b>our consciousness</b> affect the state of a particle.

- 5 years, 1 month ago

I made a extension, I hope you understand http://www.oddtheorems.com/blog/extension-on-quantum-consciousness

- 5 years, 1 month ago

I apologise for that. I'll correct it within a few hours.

Staff - 5 years, 1 month ago

"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. " Not exactly accurate. This is true only for stationary states. But impressive stuff , I wish I had your insight at 16. Very well written too.
And don't start musing on time just yet ... maybe do another one of these assignments on relativity first ?

- 5 years ago

You're correct about Stationary states. I was stupid.

Staff - 5 years ago