Relativistic Quantum Mechanics: Spin & Dirac Equation

Derives the relativistic Hamiltonian for an electron in an electromagnetic field from a covariant Lagrangian, includes Pauli spin matrices and the magnetic field coupling, and recovers the non-relativistic limit.

Derive the Dirac equation from Pauli matrices by converting the relativistic energy–momentum relation into a first-order four-component spinor equation using sigma dot del and gamma matrices.

Derive a conserved current J_mu proportional to psi_bar gamma_mu psi for the Dirac equation by combining the equation with its adjoint and using gamma_mu dagger relations, resulting in zero divergence.

Derive the non relativistic limit of the Dirac equation, showing a two-component spinor with large and small components and minimal coupling to the electromagnetic field.

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.

Explore the Dirac equation for a free particle, showing each spinor component satisfies the Klein-Gordon equation and forms a constant spinor with a free exponential, yielding the energy-momentum relation.

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 free-particle Dirac equation solutions, including spin up/down for positive and negative energies, by boosting rest-spinors to momentum and constructing u(p,s) and v(p,s) from the energy–momentum relation.

Examine how boosting along the spin quantization axis preserves sigma_z eigenstates for rest and moving frames, and derive the conserved current for positive and negative energy solutions.

Explore negative energy solutions in the Dirac equation, showing how the signs in plane-wave factors and gamma matrices with charge conjugation recast them as positive-energy states in an electromagnetic field.

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 operator K, built from gamma zero, sigma, and the total angular momentum J, commutes with the Hamiltonian and encodes the spin component along the total angular momentum.

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.

Solve the Dirac equation for hydrogen using the conserved K operator and total angular momentum, linking J, L, spin, and Sigma to obtain hydrogen eigenstates and K eigenvalues.

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.

Derive the Dirac operator relation sigma·p = (1/r) (sigma·x) (1/r) - iħ d/dr + i sigma·l, using spherical and cylindrical coordinates to solve hydrogen.

Deriving the Dirac equation for hydrogen, this lecture uses sigma dot p and angular momentum, introduces the K operator and parity, and derives coupled radial equations for f(r) and g(r).

Apply substitutions to simplify two coupled radial Dirac equations for hydrogen. Introduce dimensionless rho and the fine structure constant alpha, rewriting the system for solving.

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.

Examine hydrogen-like atoms with varying proton numbers and a single outer electron, showing discrete energy levels from the schrodinger equation and constructing the hamiltonian and kinetic energy.

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.

Derive the energy spectrum from the Schrödinger equation for a hydrogen-like system by separating variables, clarifying the potential energy in Gaussian units with epsilon zero, and outlining the hamiltonian.

Apply separation of variables to write the wave function as a product of spatial and time parts, yielding a time-dependent and a time-independent Schrödinger equation with energy as the constant.

Explore the time-independent Schrödinger equation in spherical coordinates, using the Hamiltonian with p squared over 2 mu and the hydrogen-like atom potential, and recall the Laplace in spherical coordinates.

Separate variables in the time-independent Schrödinger equation with spherical coordinates, factor the wavefunction into radial and angular parts, and derive equations featuring the l(l+1) constant.

The lecture derives the radial Schrödinger equation and explains L^2 as angular momentum; it introduces chi(r) and era variable to obtain discrete energy values and bound states.

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.

Derive a power-series solution to the radial Schrödinger equation with Q(0)=0, establish p = L+1, and use large-r behavior to infer discrete energy levels.

Derive a discrete energy spectrum by substituting and simplifying the differential equation, showing energy levels quantized by an integer n and matching hydrogen atom energy spectra, contrasting with Bohr's model.

Explore how the orbital angular momentum operator L equals r cross p yields L squared, using Levi-Civita and Kronecker deltas, and express L^2 in spherical coordinates.

Construct the gradient and Laplacian in spherical coordinates from Cartesian, define r_hat, theta_hat, phi_hat, and show the angular part L^2 depends only on theta and phi.

Create a separable solution for angular momentum using L^2 and Lz, derive the Legendre differential equation in x=cosφ with m integer, and identify Legendre functions as its solutions.

Solve the Legendre differential equation with a power-series ansatz around x=1. Express the solution as a truncated hypergeometric form, yielding Legendre polynomials and the eigenvalue l(l+1).

Derive generalized spherical harmonics by extending the solution with a (1−x^2)^{m/2} factor, relate to Legendre polynomials and hyper geometric series, and normalize psi_lm.

Examine eigenfunctions of L^2 and Lz via Y_l^m, showing L^2 ψ = ħ^2 l(l+1) ψ and Lz ψ = m ħ ψ, with L^2 eigenfunctions formed as sums over m.

Explore the Legendre polynomials P_l(x) and related P_l^m and the spherical harmonics Y_l^m, proving their orthogonality on [-1,1] and the defining differential equation.

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

Compute h_l via Rodrigues' formula by evaluating the integral of P_l(x)^2 from -1 to 1. Normalize the Legendre polynomials by sqrt(h_l) to satisfy orthonormality and the delta ll' condition.

Derive the beta gamma relationship by transforming the product gamma(x) gamma(y) into a double integral and obtaining beta(x,y) = gamma(x) gamma(y) / gamma(x+y).

Demonstrate the Legendre completeness relation by showing the normalized series over l equals the Dirac delta, using function expansion and inner products; note the proof is intuitive, not rigorous.

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.

Derive the full spherical harmonics formula and its normalization and completeness relations, using Y_l^m(Ω) and the integral inner product to reveal orthogonality and phase factors.

Explore the symmetry property of spherical harmonics and derive that Y_l^{-m}(omega) = (-1)^m [Y_l^{m}(omega)]^*, using the standard Y_l^m forms and complex conjugation.

Use completeness of spherical harmonics to expand a wave function into Y_l^m(omega) with coefficients c_lm from inner products; |c_lm|^2 gives probability of measuring L^2 eigenvalue ħ^2 l(l+1) via Legendre polynomials.

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′.

Explore the final considerations of the addition theorem for spherical harmonics, verify real-valued symmetry under complex conjugation, and connect gamma to the dot product of the normals.