Nonrelativistic Limits
A nonrelativistic description works when the particles and fields being studied vary on scales well below the rest energy. Removing the rest-energy phase reveals slow motion; choosing the appropriate energy sector and controlling the neglected terms make that approximation reliable. This chapter derives the Schrödinger and Pauli equations, calculates the leading hydrogen fine structure, and explains when a particle-only description fails.
Choose a first calculation
Section titled “Choose a first calculation”For a first pass, start with the wave equation you already know:
| Starting point | Read next | What you will calculate |
|---|---|---|
| Scalar waves | Klein–Gordon to Schrödinger | Remove the rest phase, prepare the positive-frequency data, and bound the free evolution error. |
| Spin-half particles | Dirac to Pauli | Eliminate the small component and derive the magnetic spin coupling with . |
| Hydrogenic bound states | Foldy–Wouthuysen Expansion, then Relativistic Corrections to Hydrogen | Combine kinetic, spin–orbit, and Darwin shifts and check their sum against the Dirac spectrum. |
The scalar route uses positive-frequency solutions and the Klein–Gordon norm. The spinor route uses the Dirac basis, Pauli matrices, and minimal electromagnetic coupling. Each derivation links the particular background it needs.
For a more mathematical comparison, read Nonrelativistic Limit. It specifies how states are identified and proves convergence for each fixed prepared state over a bounded time interval. Its counterexample explains why that result is not uniform over all momentum states.
Small terms can matter over long times
Section titled “Small terms can matter over long times”The momentum ratio is only one part of the approximation. Residual energies, potential-energy differences, field strengths, spatial variation, and driving frequencies must also be controlled. An absolute constant scalar-potential offset is gauge dependent; it is not by itself a physical measure of a strong field.
A small error in energy accumulates a phase error proportional to elapsed time. Likewise, a small coupling between spectral branches can produce a large transition after long resonant driving. State preparation and the observation interval therefore belong in the comparison. Antiparticle Decoupling distinguishes exact invariance under a static Hamiltonian from approximate transport of an instantaneous subspace. A one-body branch transition becomes a pair count only after specifying the quantized field and its in/out states.
Component elimination, unitary transformations, and field theory
Section titled “Component elimination, unitary transformations, and field theory”| Method | What it supplies |
|---|---|
| Effective Hamiltonians | An exact energy-dependent stationary equation, when its inverse exists, together with the reconstructed state and norm. |
| Foldy–Wouthuysen Transformation | A unitary change of representation that transforms states and observables as well as the Hamiltonian. |
| Nonrelativistic QED Preview | Low-energy field operators whose coefficients are fixed by relativistic amplitudes, including effects beyond minimal Dirac coupling. |
An eliminated stationary denominator is not automatically a local generator of time evolution. A truncated kinetic expansion is not a stable Hamiltonian at arbitrary momentum. The pages above explain the conditions needed for each use and provide explicit checks.
An electron–positron system can also be nonrelativistic: both species may move slowly and remain in the effective theory. Which modes to retain depends on the process, its accessible channels, and the accuracy sought.
References
Section titled “References”- J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
- L. L. Foldy and S. A. Wouthuysen, “On the Dirac theory of spin 1/2 particles and its non-relativistic limit,” Physical Review 78, 29–36, 1950, doi:10.1103/PhysRev.78.29.
- Richard J. Hill, Gabriel Lee, Gil Paz, and Mikhail P. Solon, “The NRQED Lagrangian at Order 1/M⁴,” Physical Review D 87, 053017, 2013, doi:10.1103/PhysRevD.87.053017.
- Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler, “Bounds for the Adiabatic Approximation with Applications to Quantum Computation,” Journal of Mathematical Physics 48, 102111, 2007, doi:10.1063/1.2798382.
- J. J. Sakurai, Advanced Quantum Mechanics, Addison–Wesley, 1967.
- Bernd Thaller, The Dirac Equation, Springer, 1992, doi:10.1007/978-3-662-02753-0.