Dirac Equation - Analytic Solution?

[LarryS]

The quantum harmonic oscillator is an analytic solution of the Schrödinger Equation. Does the original Dirac Equation for a free electron also have an analytic solution? Of course a "solution" of the Dirac Equation would consist of 4 functions.

Thanks in advance.

 

[MathematicalPhysicist]

If there were such a solution you'd definitely would find it in Wiki entry.

 

[George Jones]

Google

"Dirac equation" hydrogen

 

[dextercioby]

The harmonic oscillation is not modeled in the context of Dirac's equation, unless one leaves particle dynamics aside and does quantum field theory.

 

[fzero]

Free particle solutions are described at https://en.wikipedia.org/wiki/Dirac_spinor

 

[Avodyne]

The quantum harmonic oscillator is an analytic solution of the Schrödinger Equation.

Yes, if the potential energy is chosen to be $\frac12 k x^2$.

Does the original Dirac Equation for a free electron also have an analytic solution? Of course a "solution" of the Dirac Equation would consist of 4 functions.

Now you've switched to a "free" electron, which means that the potential energy is zero. In this case, the general solution is a plane wave times a constant spinor; see fzero's link for details.