Zitterbewegung

More from my 732-831 class from Adam Bincer…

What does the Dirac equation describe? Because of its complex vacuum structure, the answer is not single particle states in the conventional sense. For this we have two pieces of evidence (at least).

First we look at the velocity operator in the Dirac theory. Let $c=1$ and $\hbar=1$

$\mathbf{v}=\mathbf{\dot{r}}=i[H,\mathbf{r}]$

inserting the Hamiltonian

$\beta m-i\bm{\alpha}\cdot\bm{\nabla}$

we find

$\mathbf{v}=\bm{\alpha}$

however since the different components of $\bm{\alpha}$ don’t commute we see that

$[H,\mathbf{v}]\ne 0$

so that there are no simultaneous eigenstates of energy and velocity! How can this possibly describe a particle of constant energy yet non-constant velocity? We can find the actual time dependence from the Heisenberg equation of motion

$\frac{d}{dt}\mathbf{v}=i[H,\bm{\alpha}]=-2i(\bm{\nabla}+\bm{\alpha}H)=-2i(\bm{\alpha}+i\bm{\nabla}H^{-1})H$

The inverse operator of the Hamiltonian is well defined since the Hamiltonian has no zero eigenvalues, there is a symmetric gap in its spectrum $2m$ wide, centered on zero. Since $-i\bm{\nabla}=\mathbf{p}$ is a constant of the motion, as is the energy, this equation can be directly integrated

$\ln(\bm{\alpha}+i\bm{\nabla}H^{-1})|_{t_0}^t=-2iH(t-t_0)$

or (you can replace $H$ with its expectation $E$ )

$\bm{\alpha}(t)=(\frac{\mathbf{p}}{H})_t+\Big(\bm{\alpha}_0-\frac{\mathbf{p_0}}{H}\Big) \, e^{-2iH(t-t_0)}$

This oscillatory behavior is called ziitterbewegung, jittery motion, and the frequency is very high since $|H|\geq mc^2$ . What could cause such violent accelerations/oscillations of frequency $f\geq 2mc^2/\hbar$ (restoring $c,\hbar$ ) of charged particles? Sounds like pair production as the upper and lower components of the wavefunction interfere with each other. This oscillation can be remove for wavepackets made entirely of positive energy components (or negative energy) but is present for packets with both.

Our second clue comes from the so-called Klein paradox.
Consider Dirac positive energy particles incident from the left on a step barrier at the origin of height $V_0$ . To the left of the barrier we have an incident wave moving to the right

$\psi_I=e^{ikz}\left(\begin{array}{c}1\\0\\\frac{k}{E+m}\\0\end{array}\right)$

as well as a reflected wave the could conceivably have a spin flipped part

$\psi_R=ae^{-ikz}\left(\begin{array}{c}1\\0\\\frac{-k}{E+m}\\0\end{array}\right)+be^{-ikz}\left(\begin{array}{c}0\\1\\0\\\frac{k}{E+m}\end{array}\right)$

To the right of the boundary there will be the transmitted wave which also could have a spin flipped part

$\psi_T=ce^{ik'z}\left(\begin{array}{c}1\\0\\\frac{k'}{E-V_0+m}\\0\end{array}\right)+de^{ik'z}\left(\begin{array}{c}0\\1\\0\\\frac{-k'}{E-V_0+m}\end{array}\right)$

with

$k'=\sqrt{(E-V_0)^2-m^2}$

Continuity of the waves at $x=0$ show that $b=d=0$ so no spin flips occur, and that from calculation of the currents, total flux is conserved

$c=\frac{2}{1+\frac{k'}{k}\frac{E+m}{E-V_0+m}}=1+a$

$S_{trans}=S_{inc}\frac{4\frac{k'}{k}\frac{E+m}{E-V_0+m}}{(1+\frac{k'}{k}\frac{E+m}{E-V_0+m})^2}, \qquad S_{inc}=S_{trans}+S_{refl}$

But here is the strange part,

$\frac{k'}{k}\Big(\frac{E+m}{E-V_0+m}\Big)\leq 0$

$E+m < V_0$

and so under these conditions

$S_{refl} > S_{inc}$

More particles are reflected from the barrier than are incident on it! Since this only happens if the barrier height is $mc^2$ higher than the incident particle’s energy, or more, we interpret this to mean that the incident particles are stimulating particle creation inside of the barrier. This means that antiparticles must also be created to conserve all quantities known to be constants of the motion. Once again we see this because our incoming packet had all four components of the Dirac spinor.
When we solve the Schrodinger equation for a piece-wise continuous potential like $V(x)=\left\{\begin{array}{ll}0 & x<0\\ V_0 & x\ge 0\end{array}\right.$ why do we not demand that $\tfrac{d^2\psi}{dx^2}$ be continuous at $x=0$ ? Because the Schrodinger equation prescribes the value of $\tfrac{d^2\psi}{dx^2}$ everywhere to be $=-\tfrac{2m}{\hbar^2}(E-V(x))$ . The Dirac equation presecribes the values of first derivatives of the spinor wavefunction and so we make no derivative continuity demands on them.

It seems that Dirac theory does not describe single particles, but rather a many particle vacuum state with particle-like excitations. This would account for the violent oscillations in the particle velocity, of frequency just over the threshold for particle pair creation $\frac{2mc^2}{\hbar}$ .

Published by Jeff on October 30, 2021October 30, 2021

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