36.14.3The Expected Velocity and Zitterbewegung
The expected value of the velocity in a plane wave state can be simply calculated.
〈vk〉=∫
ψ†(icγ 4 γk)ψ d^3 x(icγ 4 γ 1 )u(1)~p =c
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
√
E+mc^2
2 EV
1
0
pzc
E+mc^2
(px+ipy)c
E+mc^2
(icγ 4 γ 1 )u(1)~p =c√
E+mc^2
2 EV
(px+ipy)c
E+pzmcc^2
E+mc^2
0
1
u(1)~p †(icγ 4 γ 1 )u(1)~p =E+mc^2
2 EVc(
1 0 E+pzmcc 2 (pEx+−mcipy 2 )c)
(px+ipy)c
E+pzmcc^2
E+mc^2
0
1
u(1)~p †(icγ 4 γ 1 )u(1)~p =E+mc^2
2 EVc2 pxc
E+mc^2=
pxc
EVc〈vk〉=pkc^2
EThe expected value of a component of the velocity exhibits strangebehavior when negative and
positive energy states are mixed. Sakurai (equation 3.253) computes this. Note that we use the fact
thatu(3,4)have “negative energy”.
〈vk〉=∫
ψ†(icγ 4 γk)ψ d^3 x〈vk〉=∑
~p∑^4
r=1|c~p,r|^2pkc^2
|E|+
∑
~p∑^2
r=1∑^4
r′=3mc^3
|E|[
c∗~p,r′c~p,ru
(r′)†
~p iγ^4 γku(r)
~p e− 2 i|E|t/ ̄h+c~p,r′c∗~p,ru
(r)†
~p iγ^4 γku(r′)
~p e2 i|E|t/ ̄h]
The last sum which contains the cross terms between negative and positive energy representsex-
tremely high frequency oscillations in the expected value of the velocity, known asZit-
terbewegung. The expected value of the position has similar rapid oscillations.
The Zitterbewegung again keeps electrons from being well localized ina deep potential raising the
energy of s states. Its effect is already included in our calculation asit is the source of the Darwin
term.
36.15Solution of the Dirac Equation for Hydrogen
The standard Hydrogen atom problem can be solved exactly using relativistic quantum mechanics.
The full solution is a bit long but short compared to the complete effort we made in non-relativistic