Spin

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( this is just a draft, not organized )

Spin is a intrinsics property of elementary particle, such as electron, proton, and even photon. Intrinsics means it is a built-in property, like mass, charge. Which extrinsic properties are speed, momentum.

Spin is a vector or tensor quality while charge and mass are scaler.

Spin can react with magnetic field, like charge reacts with electric field or mass react with force produce acceleration. Thus, spin is like a bar-magnet inside particle, counter part of charge.

The magnitude of spin is half integer or integer of reduced Planck’s constant \hbar . Particles with half integer of spin are classified as Fermion, and those with integer spin are Boson. they follow different statistic while interact together, thus, this creates different physics for different group.

we are not going to the mathematic description this time.

the effect of spin causes the magnetic moment, that’s why it react with magnetic field. the other thing that creates magnetic moment is angular momentum for charge particle, like electron orbiting around nucleus. So, both spin and angular momentum can be imagined as a little magnet, thus, they can interact, in physics, we call the interaction between spin and angular momentum is coupling. for example, spin-orbital coupling, spin-spin coupling, etc..

when the spin interact with external magnetic field, it will precess around the magnetic field with Larmor frequency. and the direction of the spin while undergoes procession can only be certain angle. for spin half, like electron or proton. there are only 2 directions, and we called it up and down.

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Natural unit

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on Size, Energy and Unit, we know that the speed of light better be equal to 1, that simplify the equation of relativity.

c = 1

now, we impose 1 more things, the Reduced Planck constant,  \hbar also set to 1. that simplify all equations with angular momentum or spin.

\hbar = 1

the Angular momentum:

J^2 \left| l,m \right > = j(j+1) \hbar ^2 \left| l,m\right >

J_z \left|l,m \right> = m \hbar \left|l,m\right>

now becomes :

J^2 \left| l,m \right > = j(j+1) \left| l,m\right >

J_z \left|l,m \right> = m \left|l,m\right>

when we want to calculate the real value, we can recover the \hbar by considering the dimension. the reduced Planck constant has dimension

[\hbar] = [kg][m^2][s^{-1}] = [energy][second]

for example,

E = \omega \hbar

Method

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As we know the world of nuclear physics is so small. ordinary method is not applicable to “see” this world.

we have no choice but just bombard the nucleus with electrons, protons, neutrons, etc… if we are Alice, who become much bigger then our earth and touch the moon easily. How do we understand human world? we simply pick a human, hitting on each other, see what is going on, what is the result. If we want to know how elevator work, we put a human, let him ride on it and see the result. but sometime, we will accidentally, put a car in elevator.

so, Most nuclear experiment is SCATTERING EXPERIMENT.

and the machine to conduce this kind of experiment is called Accelerator.

there is a famous quote i forget where it come from:

nuclear (particle) physics is like we figure out how a watch work by broken it and see the fragments.

The picture is really like this. we shoot particles into nucleus, and see what was knot out. how the incident particle changed. we can extract the energy change, the scatter angle, the polarization. basically are these 3 things. and using these 3 data, we construct the world of nucleus. Is it amazing???

There are some fundamental limits of the scattering experiment.

1) since the size of the nucleus is very small, the chance of hitting it is very small for 1 particle to hit another particle. thus, we use many particles hit many particles. but even doing so, only a tiny fraction of reaction takes place. most of them just pass by, say a hello. Thus, we have to create a high density particle beam, and target.

2) the particle should be moving very fast, almost same as speed of light, in order to carry enough energy to go inside the nucleus. because there is a barrier form the forces. the nucleus is something like a fortress, walled by forces. (sound like a star wars movie) another reason is, the larger the energy, the shortest distance we can probe. According to De Broglie, every particle can be treated as wave with wavelength is inversely proportional to the momentum. Thus, a faster particle has larger momentum and shorter wavelength. so, can see a smaller world. [ the De Broglie’s wavelength has some debt on weather it is a physical wave or probability amplitude, this was solved by Dirac and proved by experiment that, it means both. for more info, see discussion on “interpolation on quantum wave function” ]

This is a general property of wave. a Radio wave can easy pass though us because we are small compare to the wavelength, which is about 5 to 10 meters long. but red light can “see” us, or we can block red light, because we are much bigger. the idea is, if we want to see the detail, you have to use a smaller ruler.

3) since we are using the building block to hit another building block in investigation. they are similar size. imagine a scenario that you want to measure to speed of a car, and you use another car to hit on it, and see the bounded back car to find out the original speed. you can see, the cars hit each other and changed the original speed, and there is no way to accurate to measure to speed! same things happen in nuclear physics. the scatter particle will change to state of target, that create an uncertainty. This was formulated by Heisenberg and now called the Heisenberg’s Uncertainty Principle. which state that

change of position X change of momentum  ≥ Planck’s constant