Zitterbewegung

**Zitterbewegung** ("jittery motion" in German) is a predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of 2*mc*2*ℏ*, or approximately 1.6×1021 radians per second. For the hydrogen atom, zitterbewegung can be invoked as a heuristic way to derive the Darwin term, a small correction of the energy level of the s-orbitals.

## Theory for a free fermion

The time-dependent Dirac equation is written as

H
ψ
(

x

,
t
# )
i
ℏ

∂
ψ

∂
t

(

x

,
t
)

{\displaystyle H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) }

,
where

ℏ

{\displaystyle \hbar}

 is the (reduced) Planck constant,

ψ
(

x

,
t
)

{\displaystyle \psi(\mathbf{x},t) }

 is the wave function (bispinor) of a fermionic particle spin-½, and H is the Dirac Hamiltonian of a free particle:

# H
β
m

c
2

∑

# j
1

3

α
j

p
j

c

{\displaystyle H = \beta mc^2 + \sum\_{j = 1}^3 \alpha\_j p\_j c }

,
where

m

{\textstyle m}

 is the mass of the particle,

c

{\textstyle c}

 is the speed of light,

p
j

{\textstyle p\_j }

 is the momentum operator, and

β

{\displaystyle \beta}

 and

α
j

{\displaystyle \alpha\_j }

 are matrices related to the Gamma matrices

γ
μ

{\textstyle \gamma\_\mu }

, as

# β

γ
0

{\textstyle \beta=\gamma\_0 }

 and

α
j

=

γ
0

γ
j

{\textstyle \alpha\_j=\gamma\_0\gamma\_j }

.

In the Heisenberg picture, the time dependence of an arbitrary observable Q obeys the equation

−
i
ℏ

∂
Q

∂
t

=

[

H
,
Q

]

.

{\displaystyle -i \hbar \frac{\partial Q}{\partial t} = \left[ H , Q \right] .}

In particular, the time-dependence of the position operator is given by

ℏ

∂

x
k

(
t
)

∂
t

=
i

[

H
,

x
k

]

=
ℏ
c

α
k

{\displaystyle \hbar \frac{\partial x\_k(t)}{\partial t} = i\left[ H , x\_k \right] = \hbar c\alpha\_k }

.
where *xk*(*t*) is the position operator at time t.

The above equation shows that the operator αk can be interpreted as the k-th component of a "velocity operator". To add time-dependence to αk, one implements the Heisenberg picture, which says

α
k

(
t
# )

e

i
H
t

ℏ

α
k

e

−

i
H
t

ℏ

{\displaystyle \alpha\_k (t) = e^\frac{i H t}{\hbar}\alpha\_k e^{-\frac{i H t}{\hbar}}}

.
The time-dependence of the velocity operator is given by

ℏ

∂

α
k

(
t
)

∂
t

=
i

[

H
,

α
k

]

=
2

(

i

γ
k

m
−

σ

k
l

p
l

)

=
2
i

(

p
k

−

α
k

H

)

{\displaystyle \hbar \frac{\partial \alpha\_k(t)}{\partial t} = i\left[ H , \alpha\_k \right] = 2\left(i \gamma\_k m - \sigma\_{kl}p^l\right) = 2i\left(p\_k-\alpha\_kH\right) }

,
where

σ

k
l

≡

i
2

[

γ
k

,

γ
l

]

.

{\displaystyle \sigma\_{kl} \equiv \frac{i}{2}\left[\gamma\_k,\gamma\_l\right] .}

Now, because both pk and H are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.

First:

α
k

(
t
# )

(

α
k

(
0
)
−
c

p
k

H

−
1

)

e

−

2
i
H
t

ℏ

c

p
k

H

−
1

{\displaystyle \alpha\_k (t) = \left(\alpha\_k (0) - c p\_k H^{-1}\right) e^{-\frac{2 i H t}{\hbar}} + c p\_k H^{-1} }

,
and finally

x
k

(
t
# )

x
k

(
0
)

c
2

p
k

H

−
1

t

1
2

i
ℏ
c

H

−
1

(

α
k

(
0
)
−
c

p
k

H

−
1

)

(

e

−

2
i
H
t

ℏ

−
1

)

{\displaystyle x\_k(t) = x\_k(0) + c^2 p\_k H^{-1} t + \tfrac12 i \hbar c H^{-1} \left( \alpha\_k (0) - c p\_k H^{-1} \right) \left( e^{-\frac{2 i H t}{\hbar}} - 1 \right) }

.
The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the Compton wavelength. That oscillation term is the so-called zitterbewegung.

The zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. This can be achieved by taking a Foldy–Wouthuysen transformation. Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.

### Interpretation

In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. This can be achieved by taking a Foldy–Wouthuysen transformation. Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.

In quantum electrodynamics the negative-energy states are replaced by positron states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron pairs.

## Experimental simulation

Zitterbewegung of a free relativistic particle has never been observed directly, although there is strong evidence in favor of its existence. It has also been simulated twice in model systems that provide condensed-matter analogues of the relativistic phenomenon. The first example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation (although the physical situation is different). Then, in 2013, it was simulated in a setup with Bose–Einstein condensates.

Other proposals for condensed-matter analogues include graphene and topological insulators.