information:classification, stylistic:style_transfer, structural:expansion
Measurement of spin along an arbitrary axis

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let u = (u, u, u) be an arbitrary unit vector. Then the operator for spin in this direction is simply S u = ℏ 2 (u x σ x + u y σ y + u z σ z). {\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}

The operator S has eigenvalues of ± ħ / 2, just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x -, y -, z -axis directions.

A normalized spinor for spin- 1 / 2 in the (u, u, u) direction (which works for all spin states except spin down, where it will give 0 / 0) is 1 2 + 2 u z (1 + u z u x + i u y). {\displaystyle {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}

The above spinor is obtained in the usual way by diagonalizing the σ matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.