Relativistic particle in Coulomb potential - any analytic solution?

[petergreat]

Is there a general analytic solution to the classical motion of a relativistic charged particle in a static Coulomb potential? Of course, the non-relativistic limit is simply Kepler's problem. Quantum effects should be ignored, but relativistic effects (such as E field transforming into B field) should be fully included.

 

[Petr Mugver]

Quantum effects should be ignored, but relativistic effects (such as E field transforming into B field) should be fully included.

So you want a solution that also takes into account the action of the particle's field on the external field? (radiation reaction, Schott's term?) In this case I don't think there is an exact solution. The best you can get is a perturbative method (as far as I know). If instead you mean simply a relativistic motion (no slow motion approximation), without radiation, then look, for example, the Landau-Lifsitz book.

 

[Bob S]

If you mean bound states of the electron or other charged particles in a static central Coulomb field, look (for spin 1/2 particles) at the chapter on the relativistic solution of the hydrogen atom in Dirac Principles of Quantum Mechanics (4th Ed.) page 272, or Schiff Quantum Mechanics Eq 44.26 (page 337). For spin zero particles (e.g., pions), look at Eq 42.21 in Schiff.

Bob S

 

[petergreat]

If you mean bound states of the electron or other charged particles in a static central Coulomb field, look (for spin 1/2 particles) at the chapter on the relativistic solution of the hydrogen atom in Dirac Principles of Quantum Mechanics (4th Ed.) page 272, or Schiff Quantum Mechanics Eq 44.26 (page 337). For spin zero particles (e.g., pions), look at Eq 42.21 in Schiff.

Bob S

I'm aware that the Dirac equation for a bound electron can be solved analytically. However, I'm wondering whether analytic solutions can be found for classical motion, if we ignore radiation reaction?

 

[Petr Mugver]

I'm aware that the Dirac equation for a bound electron can be solved analytically. However, I'm wondering whether analytic solutions can be found for classical motion, if we ignore radiation reaction?

Yes, in this case a closed solution exists (I already gave you a reference, but I guess you'll find lots of articles googling), but only for the spatial trajectory of the particle, not for the time dependence (this is also the case for classical Kepler motion: we can show that trajectories are elliptic, parabolic or hyperbolic, but the time dependence is not expressible in terms of elementary functions). In the relativistic case, depending on the initial conditions (and of course if the two charges have same or opposite signs) you get a whole class of solutions: spirals that fall into the center in a finite time, elliptic orbits that show precession, etc... in any case the orbits are never closed.

 

[alxm]

I'm aware that the Dirac equation for a bound electron can be solved analytically. However, I'm wondering whether analytic solutions can be found for classical motion, if we ignore radiation reaction?

You mean something like the http://en.wikipedia.org/wiki/Old_quantum_theory#Relativistic_orbit"?

 

[George Jones]

You mean something like the http://en.wikipedia.org/wiki/Old_quantum_theory#Relativistic_orbit"?

At your link, the non-quantum stuff that petergreat wants is given under the heading Relativistic orbit, up to, but not including, "With the quantum conditions ...".

petergreat, notice that the equation of motion mathematically (but not physically, for reasons given by Petr Mugver) is the same as the equation of motion for an undamped harmonic oscillator under the influence of a constant additional force.

 

[starthaus]

Is there a general analytic solution to the classical motion of a relativistic charged particle in a static Coulomb potential? Of course, the non-relativistic limit is simply Kepler's problem. Quantum effects should be ignored, but relativistic effects (such as E field transforming into B field) should be fully included.

Analytic solutions for the "classical" case (absence of radiation breaking) exist but they are very tricky to derive. I think that what you want can be seen in the third attachment of https://www.physicsforums.com/blog.php?b=1928 blog.

 

[Dickfore]

Is there a general analytic solution to the classical motion of a relativistic charged particle in a static Coulomb potential? Of course, the non-relativistic limit is simply Kepler's problem. Quantum effects should be ignored, but relativistic effects (such as E field transforming into B field) should be fully included.

Not as far as I know. The problem is that the two-bodies are accelerated and emit electromagnetic waves. Thus, the problem becomes a coupled problem about the dynamics of two point particles and the electromagnetic field evolution that they create.

There is an approximate procedure where one can deduce a Lagrangian for a system of point charges up to terms of the order $O((u/c)^{2})$ - Darwin Lagrangian by neglecting the radiating degrees of freedom.

I don't know if the Kepler problem using this Lagrangian is analytically solvable or not.

 

[starthaus]

Yes, in this case a closed solution exists (I already gave you a reference, but I guess you'll find lots of articles googling), but only for the spatial trajectory of the particle, not for the time dependence (this is also the case for classical Kepler motion: we can show that trajectories are elliptic, parabolic or hyperbolic, but the time dependence is not expressible in terms of elementary functions). In the relativistic case, depending on the initial conditions (and of course if the two charges have same or opposite signs) you get a whole class of solutions: spirals that fall into the center in a finite time, elliptic orbits that show precession, etc... in any case the orbits are never closed.

Yes, this is correct.