Relativistic Quantum Mechanics: Spin & Dirac Equation

We derive the non relativistic limit of Dirac equation for hydrogen, yielding a Schrödinger-like equation with kinetic energy corrections, including spin-orbit coupling, Thomas precession, and the Darwin term.

Derive the Dirac equation solutions for a free particle at rest, revealing positive and negative energy spinors and spin up/down states along a quantization axis.

Derive the Dirac Hamiltonian from the equation, identify the momentum pi, and demonstrate that the Hamiltonian commutes with total angular momentum Jz by showing that [H, Lz] and [H, Sz] cancel.

Show that the commutator [K, J] vanishes, with J as the total angular momentum and Kay as its spin component, using gamma matrices and sigma operators.

Apply the Dirac equation to hydrogen with an electromagnetic field, recasting it into a Hamiltonian form and relating energy to V(r), p, and sigma matrices.

Recast the Dirac equation for hydrogen part 4 using the K operator to separate angular and radial parts and derive two coupled radial equations for g and f.

Postulate forms for F and G in the Dirac equation for hydrogen, derive recurrence relations and a determinant condition to fix allowed solutions, and select the positive s solution.

Derive the relativistic energy spectrum for hydrogen-like atoms by enforcing a terminating Dirac equation series and applying recursion relations. This yields the exact relativistic energy expression for hydrogen-like atoms.

Derive the relativistic hydrogen energy spectrum from quantum numbers of the four commuting operators and the radial equation, then recover the non relativistic limit for small Z alpha.

Explore how Rutherford's experiments sparked the shift from planetary atom models to quantum mechanics, introducing Bohr's hydrogen-like energy levels and the Schrödinger equation for a step-by-step derivation.

Describe a hydrogen-like two-particle system with electron and proton, deriving a Hamiltonian with reduced mass mu and a Coulomb-like potential V(r) = - Z e^2 / a_h.

Analyze the radial Schrödinger equation via separation of variables, relate chi(r)=e^{-r/2} q(r), and study large‑r behavior to obtain a differential equation for q and the discrete energy spectrum.

Explore the Rodrigues formula for Legendre polynomials, prove P_l(1)=1, and derive the Legendre differential equation through induction and derivative operators.

Analyze the generalized Legendre polynomials P_ml, their Rodrigo's formula, derivative relations, and the m to −m symmetry, within the differential equation with 1−x^2 terms and large-x behavior.

Show how generalized Legendre polynomials are orthogonal on [-1,1], deriving the normalization constant h_{m l} and delta_{l l'} via Rodrigo's formula and integration by parts.

Prove the addition theorem for spherical harmonics by expanding the delta distribution, using completeness, Legendre polynomials, and the angle gamma between Ω and Ω′ to relate Ω and Ω′.