Pauli spin matrices and hbar/2

[lonewolf219]

Why are the Pauli matrices multiplied by 1/2 ?? Why are they represented as σ1/2 σ2/2 and σ3/2 and not just σ1 σ2 σ3 ?

 

[The_Duck]

We want the angular momentum operators to satisfy the commutation relation

$[\hat{J}_i, \hat{J}_j] = i \epsilon_{i j k} \hat{J}_k$

For example, you can verify that the orbital angular momentum operators $\hat{\vec{x}} \times \hat{\vec{p}}$ satisfy this commutation relation. We want the spin angular momentum operators to satisfy the same commutation relation. This forces a certain normalization on the 2x2 matrices we use to represent the angular momentum operators for spin-1/2: the Pauli matrices only satisfy this commutation relation after we multiply them by 1/2. The Pauli matrices themselves satisfy a commutation relation with an extra factor of 2:

$[\sigma_i, \sigma_j] = 2 i \epsilon_{i j k} \sigma_k$.

We remove the extra factor of two by defining the spin angular momentum operators as $\hat{S}_i = \sigma_i / 2$. Then the $\hat{S}_i$ satisfy the correct commutation relations for angular momentum operators.

 

[lonewolf219]

Thanks Duck... so then the 1/2 is not related to fermions having spin of 1/2 ?

 

[wotanub]

It's because the Pauli matrices are used for spin 1/2 particles, which have intrinsic spins of either $\frac{\hbar}{2}$ or $\frac{-\hbar}{2}$.

The reason we factor out the $\frac{\hbar}{2}$ is convience. It makes the eigenvalue equation of the $\hat{S} = \frac{\hbar}{2}σ$ operators more clear

$\hat{S}_{n}\left|n\right\rangle = \frac{\hbar}{2}\left|n\right\rangle$

and it also makes it easier to evaluate rotation operators:

$\hat{R}(\phi\hat{n}) = e^{i\hat{S}_{n}\phi/\hbar}$

which need to be taylor expanded to give meaning, and will involve powers of the $\hat{S}$ operator.

 

[lonewolf219]

Wotanub, thank you... if you don't mind, maybe you could explain what the bracket notation means? I would GREATLY appreciate it! All I can see from it is that S is an operator, and maybe n is the principal quantum number? What exactly does | signify... "over"?

 

[Nugatory]

Wotanub, thank you... if you don't mind, maybe you could explain what the bracket notation means? I would GREATLY appreciate it! All I can see from it is that S is an operator, and maybe n is the principal quantum number? What exactly does | signify... "over"?

Google for "bra-ket notation".

 

[wotanub]

Wotanub, thank you... if you don't mind, maybe you could explain what the bracket notation means? I would GREATLY appreciate it! All I can see from it is that S is an operator, and maybe n is the principal quantum number? What exactly does | signify... "over"?

Sure. I highly recommend Townsend's "A Modern Approach to Quantum Mechanics" If you want to really get a hold of spin and bra-ket notation. It's the first chapter and he presents it in a very intuitive way.

$\left|ψ\right\rangle$ (called a "ket") is an expression for a state of a system. When talking only about the intrinsic spin degree of freedom, we express the state as as $\left|±n\right\rangle$ (maybe I should have used a ± in my last post) where $n$ is the axis we are measuring the spin along (ie, x, y, z or anything in between).

So for example a spin 1/2 particle that is "spin up" along the z-axis could be denoted $\left|+z\right\rangle$ and the eigenvalue equation of it with the $S_{z}$ operator would be (I dropped the hats because I don't like the way the look):

$S_{z}\left|+z\right\rangle = \frac{\hbar}{2}\left|+z\right\rangle$

and similarly for a "spin down",

$S_{z}\left|-z\right\rangle = \frac{-\hbar}{2}\left|-z\right\rangle$

since $\left|+z\right\rangle$ and $\left|-z\right\rangle$ are eigenstates of the $S_{z}$ operator.

 

[lonewolf219]

Aaaah... a light at the end of the tunnel! Thank you very much!