Quantum Mechanics Demystified 2nd Edition David Mcmahon -

7.1 Introduction In classical mechanics, angular momentum is a familiar concept: for a particle moving with momentum p at position r , the orbital angular momentum is L = r × p . In quantum mechanics, angular momentum becomes an operator, and its components do not commute. This leads to quantization, discrete eigenvalues, and the surprising property of spin – an intrinsic angular momentum with no classical analogue.

[ \sigma_x = \beginpmatrix 0 & 1 \ 1 & 0 \endpmatrix,\quad \sigma_y = \beginpmatrix 0 & -i \ i & 0 \endpmatrix,\quad \sigma_z = \beginpmatrix 1 & 0 \ 0 & -1 \endpmatrix. ] Quantum Mechanics Demystified 2nd Edition David McMahon

[ \hatL^2 |l,m\rangle = \hbar^2 l(l+1) |l,m\rangle, \quad l = 0, 1, 2, \dots ] [ \hatL_z |l,m\rangle = \hbar m |l,m\rangle, \quad m = -l, -l+1, \dots, l. ] [ \sigma_x = \beginpmatrix 0 & 1 \

[ [\hatL^2, \hatL_z] = 0. ]

These operators satisfy the fundamental commutation relations: where (\sigma_i) are the Pauli matrices:

Hence, we can find simultaneous eigenstates of ( \hatL^2 ) and ( \hatL_z ). Using ladder operators ( \hatL_\pm = \hatL_x \pm i\hatL_y ), one finds:

[ \hatS_z |+\rangle = \frac\hbar2 |+\rangle, \quad \hatS_z |-\rangle = -\frac\hbar2 |-\rangle. ] Define (\hatS_i = \frac\hbar2 \sigma_i), where (\sigma_i) are the Pauli matrices: