In the Coulomb gauge, the scalar potential φ(r,t) responds instantaneously to changes in the charge distribution. Does this mean that information can travel faster than the speed of light?
AYes — the instantaneous scalar potential directly transmits information between distant charges
BNo — the instantaneous φ is a gauge artifact; the physical fields E and B still propagate causally at c
COnly in the non-relativistic limit, where special relativity is not applicable
DNo — because the Coulomb gauge is only valid for static charge distributions
The apparent instantaneity of φ is a gauge artifact, not a physical effect. Gauge potentials are not directly observable — only E and B are measurable, and these always propagate causally at the speed of light. The instantaneous φ and the wave-equation terms in A conspire together to give the correct causal E and B fields. No physical measurement at any location can detect a change at a distant source any faster than c. This is a subtle but crucial point: the division of the electromagnetic field into potentials depends on the gauge choice; only the total physical fields are gauge-invariant and observable.
Question 2 Multiple Choice
Why is the Coulomb gauge particularly convenient for calculating atomic and molecular physics problems?
AIt makes Maxwell's equations manifestly Lorentz covariant, simplifying relativistic corrections
BIt eliminates the vector potential entirely, reducing the problem to a scalar equation
CIt separates the dominant Coulomb interaction (captured by φ) from the radiation field (captured by the transverse A), allowing perturbation theory to be structured cleanly
DIt ensures that the scalar potential is always zero outside the charge distribution
In the Coulomb gauge, ∇²φ = −ρ/ε₀, so φ is precisely the familiar Coulomb potential — the dominant electron-nucleus interaction in atomic systems. The radiation field (photon emission and absorption) lives in the transverse vector potential A (which satisfies ∇·A = 0). This clean separation lets you first compute atomic states from the Coulomb potential alone, then treat A as a perturbation responsible for transitions. Option A describes the Lorenz gauge, not Coulomb. Option B is false — A is still present and carries the radiation physics. Option D is false; the Coulomb potential extends throughout space.
Question 3 True / False
The choice of gauge (Coulomb vs. Lorenz) changes the values of the physically measurable electric and magnetic fields at a given point.
TTrue
FFalse
Answer: False
False. This is the fundamental point of gauge invariance. The electric field E = −∇φ − ∂A/∂t and magnetic field B = ∇×A are unchanged by a gauge transformation (adding ∇λ to A and subtracting ∂λ/∂t from φ). Different gauges assign different values to the potentials φ and A, but these are mathematical conveniences — only E and B have direct physical meaning. Any gauge gives the same predictions for observable quantities; the choice of gauge is purely a matter of computational convenience.
Question 4 True / False
The Lorenz gauge is preferred over the Coulomb gauge for non-relativistic atomic physics because it produces simpler equations for atomic energy levels.
TTrue
FFalse
Answer: False
False — the preference is reversed. The Coulomb gauge is preferred for non-relativistic atomic physics precisely because it separates Coulomb interactions from radiation effects in a way that matches the dominant physics: the electron-nucleus Coulomb interaction is much larger than the radiation corrections, making Coulomb gauge a natural starting point for perturbation theory. The Lorenz gauge is preferred for relativistic calculations because it treats space and time symmetrically and keeps Lorentz covariance manifest — a priority in quantum field theory and high-energy physics, not in atomic spectroscopy.
Question 5 Short Answer
Explain why the apparently instantaneous scalar potential in the Coulomb gauge does not violate the principle of causality.
Think about your answer, then reveal below.
Model answer: The scalar potential φ is not directly observable. Physical measurements detect only the electric and magnetic fields E and B, which are gauge-invariant combinations of φ and A. In the Coulomb gauge, φ is instantaneous but A satisfies a wave equation with a retarded source term; together they produce E and B that propagate causally at c. The instantaneity of φ is a mathematical artifact of the gauge choice — it cancels exactly with terms in A to give the correct causal fields. No experiment can detect the 'instantaneous' change in φ directly.
This subtlety is important for understanding what gauge choice represents. A gauge transformation changes φ and A but leaves E and B unchanged. Since only E and B are observable, you can always rewrite the problem in a different gauge without changing any physical prediction. The Coulomb gauge happens to assign an instantaneous character to φ, but this is a feature of how the math is organized, not a feature of the physics.