Zitterbewegung
Encyclopedia
Zitterbewegung is a theoretical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation
. 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 (at the speed of light) of the position of an electron around the median, with a circular frequency of
, or approximately 1.6 Hz
.
Zitterbewegung of a free relativistic particle has never been observed, but the behavior of such a particle has been simulated with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different).
where
is the Dirac Hamiltonian
for an electron in free space
in the Heisenberg picture
implies that any operator Q obeys the equation
In particular, the time-dependence of the position operator
is given by
where
.
The above equation shows that the operator
can be interpreted as the kth component of a "velocity operator".
The time-dependence of the velocity operator is given by
where
.
Now, because both
and 
are time-independent, the above equation can easily be integrated twice to
find the explicit time-dependence of the position operator. First:
Then:
where
is the position operator at time 
.
The resulting expression consists of an initial position, a motion proportional to time, and an unexpected oscillation term with an amplitude equal to the Compton wavelength
. That oscillation term is the so-called "Zitterbewegung".
Interestingly, 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.
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
. The existence of such motion was first proposed by Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
in 1930 as a result of his analysis of the wave packet
Wave packet
In physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...
solutions of the Dirac equation for relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
electrons in free space, in which an interference
Interference
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have...
between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with a circular frequency of
, or approximately 1.6 HzHertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
.
Zitterbewegung of a free relativistic particle has never been observed, but the behavior of such a particle has been simulated with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different).
Theory
The time-dependent Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
where
is the Dirac HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
for an electron in free space
in the Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
implies that any operator Q obeys the equation
In particular, the time-dependence of the position operator
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...
is given by
where
.The above equation shows that the operator
can be interpreted as the kth component of a "velocity operator".The time-dependence of the velocity operator is given by
where
.Now, because both
and are time-independent, the above equation can easily be integrated twice tofind the explicit time-dependence of the position operator. First:
Then:
where
is the position operator at time .The resulting expression consists of an initial position, a motion proportional to time, and an unexpected oscillation term with an amplitude equal to the Compton wavelength
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
. That oscillation term is the so-called "Zitterbewegung".
Interestingly, 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
Foldy-Wouthuysen transformation
The Foldy-Wouthuysen transformation is a unitary transformation on a fermion wave function of the form:\psi \to \psi '=U\psi where the unitary operator is the 4x4 matrix:...
. Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused by
interference between positive- and negative-energy wave components.
See also
- Casimir effectCasimir effectIn quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
- Lamb shift
- Stochastic electrodynamicsStochastic electrodynamicsIn theoretical physics, Stochastic Electrodynamics is a variant of Classical Electrodynamics which posits the existence of a classical Lorentz Invariant radiation field having statistical properties similar to that of the electromagnetic zero-point field of Quantum Electrodynamics...
: Zitterbewegung is explained as an interaction of a classical particle with the zero-point field.
Further reading
- E. Schrödinger, Über die kräftefreie Bewegung in der relativistischen Quantenmechanik ("On the free movement in relativistic quantum mechanics"), Berliner Ber., pp. 418–428 (1930); Zur Quantendynamik des Elektrons, Berliner Ber, pp. 63–72 (1931)
- A. Messiah, Quantum Mechanics Volume II, Chapter XX, Section 37, pp. 950–952 (1962)
External links
- The Zitterbewegung Interpretation of Quantum Mechanics, an alternative explanation in addition to positive-negative energy states interference.
- Zitterbewegung in New Scientist
- Geometric Algebra in Quantum Mechanics
- Summary of trapped ion simulation