3.1 Angular Momentum Operators and Commutation Relations
📚 Angular Momentum Algebra
Angular momentum operator: \(\hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}\)
Fundamental commutation relations:
\[ [\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z, \quad [\hat{L}_y, \hat{L}_z] = i\hbar\hat{L}_x, \quad [\hat{L}_z, \hat{L}_x] = i\hbar\hat{L}_y \]
Ladder operators:
\[ \hat{L}_\pm = \hat{L}_x \pm i\hat{L}_y \]
\[ \hat{L}_\pm |l,m\rangle = \hbar\sqrt{l(l+1) - m(m\pm 1)} |l, m\pm 1\rangle \]
Eigenvalues:
\[ \hat{L}^2 |l,m\rangle = \hbar^2 l(l+1) |l,m\rangle, \quad \hat{L}_z |l,m\rangle = \hbar m |l,m\rangle \]
Summary
In this chapter, we studied angular momentum and spin. The angular momentum algebra is characterized by fundamental commutation relations, ladder operators, and quantized eigenvalues. We examined spin 1/2 systems described by Pauli matrices, the Bloch sphere representation, and measurement probabilities. Finally, angular momentum addition introduces Clebsch-Gordan coefficients and the distinction between triplet and singlet states.
In the next chapter, we will study perturbation theory and scattering theory.