Roger's Equations Blog

### Roger's Equations

This blog is all about science and technology (with occasional math thrown in for fun). The goal of this blog is to try and pass on the sense of excitement and wonder I feel when I read about these topics. I hope you enjoy the posts.

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# Fermions and Bosons

Posted July 09, 2007 5:08 PM by Bayes

All fundamental particles in nature can be categorized as either a fermion or a boson.

Bosons have integral spin and can occupy the same "state". All of the force carriers (Photons, Gluons, etc.) and mesons (Pions, Kaons, etc.) are bosons.

Fermions have half integral spin and cannot occupy the same state (Pauli Exclusion Principle). Quarks (up, down, strange, etc.) and Leptons (Electrons, Muons, etc.), as well baryons (Protons, Neutrons, etc.) are all Fermions.

Proton Example:

A proton is a Baryon. Baryons by definition are made of three quarks, which are fermions. A proton is made up of two up quarks and a down quark (uud). Each quark is a fermion, so a combination of three quarks is a fermion as well. So a proton is a fermion.

So where do Fermions and Bosons come from? Why are there two types of fundamental particles? Is there a quantum mechanical explanation?

The Importance of Being Identical

In probability, things that are identical lead to different results than things that aren't. For instance, say you have a bag with two marbles. If the marbles are different colors, say red and blue, then the odds of pulling out a specific color, say blue, is 50%. However if both marbles are blue, in other words identical (in terms of color), then the probability of pulling out a blue marble is 100%.

I know that seems pretty obvious, but it's important to see that distinction because it's about to become much less clear. Classically if you have two identical things, you can tell them apart by their position. You could say, blue marble A is to the left of blue marble B. You could also tell them apart if you knew their momentum. You could say blue marble A has larger momentum than blue marble B.

So here's the problem, in Quantum Mechanics you don't know where a particle is exactly, nor its momentum exactly, so you're suddenly out of tricks to tell identical particles apart.

The Solution

In quantum mechanics, we don't talk about individual particles, we talk about the state of a system which could be 1 particle or many. This is because of the uncertainty principle (Here is an earlier blog entry where I talk about this). For a two particle system, the state can be defined as:

where r12=x12+y12+z12 and r22=x22+y22+z22 with r1 corresponding to particle number 1 and r2 particle number 2.

Thus to calculate the probability of the wavefunction we write:

|ψ(r1,r2)|2

But since the particles are identical, we can switch the particles and we get the same probability as before which means:

|ψ(r1,r2)|2 = |ψ(r2,r1)|2

Solving the equation above we get, we get

ψ(r1,r2) = ± ψ(r2,r1)

So what does it mean? Well it turns out it means something very, very important. It says that there are two different types of two identical particles. One in which switching the two particles results in no change in the wavefunction (symmetric) and one in which switching the particles adds a negative sign to the wavefunction (antisymmetric). So how can we represent these two different types of wavefunctions for identical particles? Lets try:

ψS = 1/√2(ψ(r1,r2) + ψ(r2,r1))
ψA = 1/√2(ψ(r1,r2) - ψ(r2,r1))

We can see that ψS = 0 when ψ(r1,r2) = - ψ(r2,r1), but ψS exists when ψ(r1,r2) = ψ(r2,r1). Also you can see for yourself that ψA = 0 when ψ(r1,r2) = ψ(r2,r1), and exists only when ψ(r1,r2) = - ψ(r2,r1) . Thus ψS is representative of the symmetric solution for identical particles and ψA is representative of the antisymmetric solution for identical particles.

Bosons and Fermions - What's in a name?

So now we know that if you have two identical particles, there are two types of wavefunctions which can describe the system. One is symmetric and the other is antisymmetric, but aside from being an interesting mathematical result, aren't we taking it a little too seriously? You'll calculate the same probability for both wavefunctions, which is what is what is measurable anyway. Wavefunctions are just abstractions, who cares.

Well, it turns out that it matters, a lot. Lets take a closer look at the symmetric and antisymmetric wavefunctions and consider a special case to understand why:

Here are the wavefunctions for symmetric and antisymmetric wavefunctions:

ψS = 1/√2(ψ(r1,r2) + ψ(r2,r1))
ψA = 1/√2(ψ(r1,r2) - ψ(r2,r1))

So what happens if r1=r2, in other words, the particles are in the same position (for those of you who object, relax, this is just a particular example of the more general "state").

ψS = 1/√2(ψ(r1,r1) + ψ(r1,r1)) = 2/√2 ψ(r1,r1) = 2 ψ(r1,r1)
ψA = 1/√2(ψ(r1,r1) - ψ(r1,r1)) = 0

So what does this mean?

Well, lets remember that the modulus squared (|ψ2|) of a wavefunction is related to the probability. Lets also remember that |02| = 0. Lastly lets remember that there is a law called the Conservation of Probability which says that probability cannot be created or destroyed.

So lets say you've got two identical particles in the universe, described by a nonzero state. In the symmetric case:

ψS = 1/√2(ψ(r1,r2) + ψ(r2,r1))

At r1=r2 (when the two particles are at the same spot), the probability is nonzero and conserved, everythings ok.

However,

In the antisymmetric case:

ψA = 1/√2(ψ(r1,r2) - ψ(r2,r1))

At r1=r2, (when the two particles are in the same spot), the probability goes to zero, which means that probability is not conserved. Since conservation of probability is a law, that means only one thing, identical antisymmetric particles can't ever exist at the same location (or more generally "in the same state") at the same time. This is called the Pauli exclusion principle.

Location, Location, Location - How the statistics are effected

Because of this unique difference between the ψA and ψS , there are dire consequences to the statistics that arise when the number of particles increase. ψS follow Bose-Einstein statistics which is why they are called Bosons. ψA follow Fermi-Dirac statistics and thus are called Fermions. Fermions follow the Pauli Exclusion Principle which says that two identical fermions cannot occupy the same state.

Certain Uncertainty

So there you have it. The uncertainty principle creates a situation that can only be solved by two types of fundamental particles, fermions and bosons. Those particle types follow specific statistics that gives rise to matter and forces. If there wasn't uncertainty, then we would be able to distinguish between fundamental particles and the antisymmetric two particle state, fermions, wouldn't exist. No electrons, protons, neutrons, and countless other particles. It's also worth mentioning that if there was only one particle, none of these issues come up, since the properties of Fermions and Bosons are only relevant when there are more than one particle. The addition of a second particle has led to symmetry breaking.

That's all for now. Remember if you'd like something derived or would like to have a subject covered in this blog, please email me your suggestions.

Roger

Special thanks to Wikipedia and Quantum Mechanics by B.K. Agarwal and Hari Prakash.

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#1

### Re: Fermions and Bosons

07/11/2007 1:00 PM

Interesting. Thank you. Note that more information may be needed to identify particals, marbles, etc. In your "The Importance of Being Identical", you state: "You could say, blue marble A is to the left of blue marble B". But if the observers are facing each other, the statement "to the left of" will not be true. Sorry to be picayune, but the statement just caught my attention. With best regards.

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#2

### Re: Fermions and Bosons

07/11/2007 1:14 PM

Thanks for your comment. The example you provide of the two people facing each other illustrates that a choice of coordinate system must be made in order to assign postion (of anything). The important point is that in classical physics, position can be defined exactly, so identical particles can be distinguished by their relative positions to one another. However in quantum mechanics, due to the uncertainty principle, this way of differentiating the particles by position is no longer possible, which results in two different types of indentical particles (Fermions and Bosons).

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#4

### Re: Fermions and Bosons

07/13/2007 1:17 AM

Hi roger,

I have been contemplating my own theory and everything you have said goes toward my point. I believe fermiones and bosons ARE identical. The reason they behave differently is because one is inverted to the other. That is to say, their external force is opposite, yet, they are one and the same thing. A fermione is a boson, only inverted. something like that.

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#5

### Re: Fermions and Bosons

07/13/2007 2:30 PM

Hi Dave (is Dave alright, or do you prefer something else?),

Thanks for your interest. I've been fascinated with the fundamenatal particle picture of the Standard Model for a while now. The thing is so damned complicated and only seems to be getting worse, however, it does a fairly nice job describing the world we live in. I think you're expressing something a lot of us feel, that there must be something more basic underneath that's producing these issues.

You Said "A Fermion is a Boson, only inverted".

When you say "inverted", in what sense do you mean?

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#6

### Re: Fermions and Bosons

07/16/2007 2:23 AM

its my own theory, but I believe the everything is made of an infinite number of up to infinitely small electromagnetic fields, from something on the size order of strings in string theory, to the largest atomic shells. nothing more. Spherical in shape, with the positive field on the out side and negative field on the inside or visa versa. I believe they can "invert" themselves upon increases or decreases in energy, making the positive field invert to the interior and the negative field invert to the exterior. Thus, identical particles can behave differently.

When I look at everything from this point of view, everything seems to make sense to me.

BYW, thank you very much for your writings. I am extremely fascinated by all of it.

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#7

### Re: Fermions and Bosons

07/16/2007 8:55 AM

David,

That is an unusual theory. Have you tried to model it mathematically? In other words, do you have any equations based on your idea?

Roger

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#8

### Re: Fermions and Bosons

07/16/2007 9:43 PM

Roger,

No, I haven't done very much of the math yet. It gets so complicated that I'm not even sure I can do all the math. I have come to this belief mostly through much reading, here and many other places, through general observations throughout my lifetime, and through some unique experiments I have completed. I do have a few basic equations, one which would help explain and predict quantum particle behavior, and some others. I am still working on several experiments and have only recently started working on the equations.

I don't work in the field and don't have too much time to spend on it so the process is slow, but every free minute I do have, I either read or experiment.

I can verbally explain it, and apply that understanding to the real world in any situation, and it seems to fit. I'll keep working on it.

David

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#9

### Re: Fermions and Bosons

07/17/2007 1:51 AM

David,

Well, I wish you luck. Anything short of a mathematical explanation is insufficient in physics so you certainly want to work on that. There are thousands of people with ideas, some of which may be right, but if they can't express them as formulas that can be checked, those ideas aren't very helpful. Math is a discipline that requires commitment and hard work but can be very rewarding if you're willing to make the sacrifice.

Best Regards,

Roger

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#3

### Re: Fermions and Bosons

07/12/2007 3:28 PM

Does this mean that Superman can be in the same place as his twin, but not in the same place as Bizarro (the antisymmetric case)?

Sorry.

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#10

### Re: Fermions and Bosons

10/08/2008 4:39 AM

hey roger, so are the protons used in the LHC symetric or anti-symetric? if they are anti-symetric and can not exist in the same place and we try and collide them at near speed of light... is it breaking, bending, or finding a loophole in our laws of physics...

since you love math so much, do you know equations to determine the effect (if any) the CMBR has on your bosons and fermions. i suspect it has a much larger role in physics then many beleive, time will tell. perhaps even acting as time itself, since it is old light from the "big bang" echoing through the cosmos.

my math is rusty, and calculating time is... time consuming. not a theory i can come close to backing up yet. all in due time, so long as the speed of light remains a constant.

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#11

### Re: Fermions and Bosons

10/08/2008 5:00 AM

since you love math so much, do you know equations to determine the effect (if any) the CMBR has on your bosons and fermions. i suspect it has a much larger role in physics then many beleive, time will tell. perhaps even acting as time itself, since it is old light from the "big bang" echoing through the cosmos.

Just a question, Why does everyone believe the big bang theory to be true, when it hasn't been proven and can't be? It's a great story that has gotten alot of funding for researchers, but it simply couldn't have happened.

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#12

### Re: Fermions and Bosons

10/08/2008 6:18 AM

hence the parenthesis around "Big Bang". CMBR exists, and it is as old as time itself. i honestly could care less if were in a static universe, expanding, or shrinking cosmos. I could absolutely give a flying monkey out of my arse if god created the earth, that story is more of a fictional dillusion then santa clause and the tooth fairy combined with OJ simpsons innocent verdict.

My question was about the effect of a particle effecting other particles and time space.

dont get hung up on the wrong part of the question

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#13

### Re: Fermions and Bosons

05/12/2009 2:32 PM

Have you ever worked with gravitation, astrophysics, dark matter, dark energy?

Would you be interested in discussion and possible attempt to derive a new equation to explain them?

rt

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#14

### Re: Fermions and Bosons

05/13/2009 1:06 PM

I have been working on one recently and would be very interested. please contact me directly at davidarheault@yahoo.com

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