130_notes.dvi

and compare it to the original derivative.

v ̇j(t) =

i

̄h

(− 2 Evj+ 2c^2 pj) =

i

̄h

(− 2 E(c^2 pj/E+ (vj(0)−c^2 pj/E)e−^2 iEt/ ̄h) + 2c^2 pj)

=

i

̄h

((− 2 c^2 pj+ (− 2 Evj(0) + 2c^2 pj)e−^2 iEt/ ̄h) + 2c^2 pj) =

i

̄h

(− 2 Evj(0) + 2c^2 pj)e−^2 iEt/ ̄h

This checks so the solution for the velocity as a function of time is correct.

vj(t) =c^2 pj/E+ (vj(0)−c^2 pj/E)e−^2 iEt/h ̄

There is a steady motion in the direction of the momentum with the correct magnitudeβc. There
are also very rapid oscillations with some amplitude. Since the energy includesmc^2 , the period of

these oscillations is at most 22 mcπh ̄ 2 =mcπhc ̄ (^2) c=(3.14)(197 0. 5 MeV.^3 MeV F(c) )= 1200F/c=^1200 ×^10
− 13
3 × 1010 = 4×^10
− 21
seconds. This very rapid oscillation is known as Zitterbewegung. Obviously, we would see the same
kind of oscillation in the position if we integrate the above solution for the velocity. This very rapid
motion of the electron means we cannot localize the electron extremely well and gives rise to the
Darwin term. This operator analysis is not sufficient to fully understand the effect of Zitterbewegung
but it illustrates the behavior.

36.14.2Expansion of a State in Plane Waves

To show how the negative energy states play a role in Zitterbewegung, it is convenient to go back to
the Schr ̈odinger representation and expand an arbitrary statein terms of plane waves. As with non-
relativistic quantum mechanics, the (free particle)definite momentum states form a complete
setand we can expand any state in terms of them.

ψ(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/h ̄

Ther= 1,2 terms are positive energy plane waves and ther= 3,4 states are “negative energy”.
The differing signs of the energy in the time behavior will give rise to rapid oscillations.

The plane waves can be purely either positive or “negative energy”,however,localized states have
uncertainty in the momentum and tend to have both positive and “negative energy”
components. As the momentum components become relativistic, the “negative energy” amplitude
becomes appreciable.
c 3 , 4
c 1 , 2

≈

pc

E+mc^2

Even the Hydrogen bound states have small “negative energy” components.

The cross terms between positive and “negative energy” will give rise to very rapid oscillation of
the expected values of both velocity and position. The amplitude of the oscillations is small for
non-relativistic electrons but grows with momentum (or with localization).