# What is electron spin and intrinsic spin of elementary particles?

#### anantchowdhary

What exactly is electron spin and intrinsic spin of elementary particles.Any link teaching these concepts from scratch would also be appreciated .

Thnx

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#### anantchowdhary

Sorry still havent understood at all!

What is the quantum number?

#### Crosson

All of the quantum mechanics of spin originates with the operator for angular momentum (remember the components of angular momentum come from a cross product p x r).

#### jtbell

Mentor
Sorry still havent understood at all!
OK, let's go backwards a bit. How much do you know about angular momentum in classical mechanics? Do you know about how to represent angular momentum as a vector?

#### anantchowdhary

I just about know that L=r X p

where p is linear momentum and r is the displacement position vector

#### jtbell

Mentor
OK, that's what we use for orbital angular momentum, like the earth going around the sun. The cross product gives you a vector that's perpendicular to the plane of the orbit. In QM, the vector $\vec L$ is quantized, both in magnitude and in direction. The magnitude is quantized according to

$$| \vec L | = L = \sqrt{l(l+1)} \hbar$$

where $l$ is an integer 0, 1, 2, 3... Furthermore, the component of $\vec L$ along any direction is quantized. Usually we talk about the z-direction but it can actually be any direction. After you've chosen the value of $l$, then

$$L_z = m_l \hbar$$

where $m_l$ can have values ranging from $-l$ to $+l$ in steps of 1. For example, if $l = 2$, then the possible values of $m_l$ are -2, -1, 0, +1, +2.

Something like the earth also has spin angular momentum, from spinning around its own axis. We can describe this with a vector $\vec S$ that points along the axis of rotation. Even though particles like electrons actually aren't little tiny spinning balls, they still have intrinsic angular momentum which we often call "spin," and we use the vector $\vec S$ for it.

The rules for quantizing $\vec S$ are similar to the rules for quantizing $\vec L$:

$$| \vec S | = S = \sqrt{s(s+1)} \hbar$$

$$S_z = m_s \hbar$$

where $m_s$ can have values ranging from $-s$ to $+s$ in steps of 1. The differences from orbital angular momentum are:

1. $l$ must be an integer, but $s$ can be either integer or half-integer.

2. For a particular particle (e.g. electron) $l$ and/or $m_l$ can change when its "orbit" changes, but $s$ is always the same for a particular kind of particle. For example, all electrons have $s = 1/2$, and so they must have $m_s = -1/2$ ("spin down") or $m_s = +1/2$ ("spin up").

However, $m_s$ can change. With the right circumstances, we can "flip" an electron's spin from $m_s = -1/2$ to $m_s = +1/2$ or vice versa.

#### anantchowdhary

sry im not clear with the equation $$| \vec L | = L = \sqrt{l(l+1)} \hbar$$

What do you mean by quantizing $\vec L$ ?

The help was much appreciated.Thnx

#### jtbell

Mentor
The magnitude of $\vec L$ (which we usually write as just $L$) can have only certain specific values, corresponding to the allowed values of $l$. For example, $l = 0$ gives $L = 0$; $l = 1$ gives $L = \sqrt{2} \hbar$; $l = 2$ gives $L = \sqrt{6} \hbar$; etc.

#### Schrodinger's Dog

That seems fairly easy to follow, what is the difference with anti matter particles like a positron, do they spin in the opposite direction ie they go towards the other half of the circle? Or do they just spin in an opposite direction, starting at an opposite position, sorry if this is a stupid question.

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#### jtbell

Mentor
No, spin is not one of the things that's "opposite" for an antiparticle. For example, an electron can already have its spin either "up" or "down" along a given axis, and likewise for a positron.

#### anantchowdhary

I just would like to know how the equation: $$| \vec L | = L = \sqrt{l(l+1)} \hbar$$ came into being.

and why is the spin defined in such a way?

#### Schrodinger's Dog

No, spin is not one of the things that's "opposite" for an antiparticle. For example, an electron can already have its spin either "up" or "down" along a given axis, and likewise for a positron.
I see, so it's just the charge that is reversed? e+ e- the spin is the same or can be the same according to an axis, thanks.

#### anantchowdhary

Why is the spin defined in such a way?How do we get the equation?

#### Hootenanny

Staff Emeritus
Gold Member
Why is the spin defined in such a way?How do we get the equation?
Orbital angular momentum falls out when the Schrödinger Wave Equation is solved for a Hydrogen atom.

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