 ### 8. The Dirac Equation

In this chapter, we obtain the Dirac equation with external fields in three dimensions (Section 8.1). The number of components of the Dirac spinors is doubled compared to the one-dimensional situation. Roughly speaking, there are two spin components for each sign of the energy.

In the presence of external fields, the Dirac equation cannot be invariant under Lorentz transformations. But the Dirac equation is covariant in the sense that a Lorentz-transformed solution of the Dirac equation is a solution of the Dirac equation with an appropriately transformed potential energy (Section 8.2). It is possible to classify the potential functions according to their behavior under Poincaré transformations as scalar, electromagnetic, and tensor fields (Section 8.3).

Wave packets with negative energy behave quite differently from wave packets with positive energy if put into an external electromagnetic field. By introducing the operation of charge transformation, one can see that a wave packet with negative energy actually describes the behavior of a particle with positive energy but with opposite charge (Section 8.4). We can thus interpret the solutions with negative energy as antiparticles. But this interpretation only works in situations where the splitting of the Hilbert space in electronic and positronic states is meaningful and unambiguous. A counter-example showing the limits of Dirac theory is the Klein paradox, where particles starting as electrons may end up as positrons.

In Section 8.5, we investigate the connection between Dirac's theory and the nonrelativistic Pauli equation for particles with spin. The eigenvalues and eigenfunctions of the Dirac equation tend to their nonrelativistic counterparts as c goes to infinity. We derive some formulas that let us compute relativistic perturbations of nonrelativistic energies up to first order in 1/c2.

The role of spherical symmetry in relativistic quantum mechanics is as important as in nonrelativistic quantum mechanics. In Section 8.6, we describe the angular-momentum subspaces of Dirac's theory. The radial Dirac equation becomes a system of two ordinary differential equations.

The hydrogen atom is perhaps the most important testing ground for any quantum mechanical theory.
Fortunately, the Dirac equation for a hydrogen-like system can be solved analytically, and the results are in almost perfect agreement with the measurements. This success was one of the main reasons for the quick acceptance of the Dirac equation. In Section 8.8, we solve the radial Dirac equation for the hydrogen atom by factorization methods ("supersymmetry"). The higher symmetry of the Coulomb problem in the relativistic case is related to the conservation of the Biedenharn-Johnson-Lippmann operator.

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• 8.1. The Dirac Equation 378
• 8.1.1. Free Dirac operator in two and three dimensions 378
• 8.1.2. Properties of the free-particle Dirac operator 380
• 8.1.3. Electromagnetic fields 381
• 8.2. Relativistic Covariance 382
• 8.2.1. Poincaré transformations 382
• 8.2.2. Poincaré covariance of the Dirac equation 384
• 8.2.3. Velocity transformations 386
• 8.2.4. Rotations 387
• 8.2.5. Unitary representation 388
• 8.3. Classification of External Fields 389
• 8.3.1. Poincaré transformations of external fields 389
• 8.3.2. Electromagnetic vector potential 390
• 8.3.3. Scalar potential 391
• 8.3.4. Anomalous magnetic moment 391
• 8.4. Positive and Negative Energies 392
• 8.4.1. Negative-energy wave packets are positrons 392
• 8.4.2. Relativistic scattering in one dimension: The Klein paradox 395
• 8.4.3. Physical Hilbert space and relativistic observables 398
• 8.4.4. Relativistic confinement 399
• 8.5. Nonrelativistic Limit and Relativistic Corrections 402
• 8.5.1. The nonrelativistic limit 402
• 8.5.2. Relativistic corrections 404
• 8.5.3. Spin-orbit interaction and the Darwin term 407
• 8.6. Spherical Symmetry 410
• 8.6.1. Matrix potentials with spherical symmetry 410
• 8.6.2. Operators that commute with the Dirac operator 411
• 8.6.3. Angular-momentum eigenfunctions 413
• 8.6.4. The angular-momentum subspaces 414
• 8.7. The Dirac-Coulomb Problem 417
• 8.7.1. The radial Dirac-Coulomb equation 417
• 8.7.2. A useful similarity transformation 418
• 8.7.3. The second-order equations 419
• 8.7.4. Supersymmetry 420
• 8.7.5. The ground state 422
• 8.7.6. The first excited state 424
• 8.7.7. Further eigenfunctions 425
• 8.8. Relativistic Hydrogen Atom 426
• 8.8.1. Eigenvalues and eigenfunctions 426
• 8.8.2. Degeneracy and higher symmetry 428
• 8.8.3. Angular momentum and spectroscopical notation 428
• 8.8.4. Fall to the center 430