Spin (physics)

In physics, spin is an intrinsic angular momentum associated with microscopic particles. It is a purely quantum mechanical phenomenon without any analogy in classical mechanics. Whereas classical angular momentum arises from the rotation of an extended object, spin is not associated with any rotating internal masses, but is intrinsic to the particle itself. Elementary particles such as the electron can have non-zero spin, even though they are believed to be point particles possessing no internal structure. The concept of spin was introduced in 1925 by Ralph Kronig, and independently by George Uhlenbeck and Samuel Goudsmit.

Mathematical formulation of Spin ½

The spin operator S behaves very much like L (angular momentum) where l = ½.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The spin operator has two eigenvalues: pm rac{hbar}{2} , which corresponds for two eigenstates — spin up and spin down.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It is common to measure the spin in one direction — and the corresponding operator will be mathbf{S} cdot hat{n} where n is a unit vector in the desired direction and

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mathbf{S} = rac{hbar}{2} mathbf{sigma} = rac{hbar}{2} left( sigma _x hat{x} + sigma _y hat{y} + sigma _z hat{z} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

is the vectorial spin operator and the σ-s are Pauli matrices.

Related Topics:
Vectorial - Pauli matrices

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For example: Let's measure the spin in the z direction (in cartesian coordinates). Then Sz has two eigenstates — spin up and spin down. If we assign coordinates vectors as follow

Related Topics:
Cartesian coordinates - Coordinates vector

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: | {uparrow} angle = left ert {m = +rac 1 2} ight angle = egin{bmatrix} 1 \ 0 end{bmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: | {downarrow} ang = left ert {m = -rac 1 2} ight ang = egin{bmatrix} 0 \ 1 end{bmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

then the corresponding operator in that representation will be

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: S_z = rac{hbar}{2} sigma _z = rac{hbar}{2} egin{pmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1&0\ 0&-1 end{pmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In the matrix representation, the operator acts on the coordinate vectors, often called "Spinors".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~