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Zitterbewegung, the trembling motion of a moving electron |
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P.A.M. Dirac was very proud of the equation he derived for an electron; so much so he had it engraved in simplified form on his tombstone.
In less abbreviated form the Dirac equation is
The symbols displayed in red in the above and in the following are matrices of larger dimensions than 1×1. (Remember that a vector is a particular kind of matrix.) The symbol A denotes a vector of matrices to be specified later. The symbol P denotes a vector of the following form
where m is the mass of the electron and p_{j} is the momentum along the j-th axis. The
operator for P is found by substituting −ih(∂/∂x_{j} for p_{j}.
Thus
The function H_{D} is known as the Dirac Hamiltonian.
In the Heisenberg Picture of Quantum Theory the operators may have a time dependence but the state variables are time independent.
In the Heisenberg Representation of quantum theory any operator Q is governed by the equation
where the bracket symbol is defined as
Let X be the particle position vector (x_{1}, x_{2}, x_{3},). Now consider
Since (∂x_{j}/∂x_{k})=δ_{jk} when AP^{^} operates on x_{j} the only nonzero term is
On the other hand since α_{j} is a constant x_{j}AP^{^} has no nonzero terms. Thus
Combining all such terms gives
Thus cA is in the nature of a velocity matrix.
The velocity operator is obtained in the Heisenberg picture by defining
The time-dependence of this velocity operator is given by
Elsewhere it shown that
where I is the appropriate identity matrix.
This means that
Thus
The above formula is confirmed by a mathermatical manipulation. The term [H_{D}, A^] can be expressed as
In Albert Messiah's Quantum Mechanics it is shown that
For a free electron the parameters P and H_{D} are constants and the above equation can be integrated with respect to time to give
where W is equal to
(A(0) − (c/H_{D})P)
and the frequency ν is
2H_{D}/h. The minimum value of H_{D} would be the rest mass energy of an
electron, wich is 8.187×10^{-14} joules. Thus the minimum frequency for the zitterbewegung
is 2*8.187×10^{-14}/1.0546×10^{-34}=1.5526×10^{21} per second.
So velocity is equal to a constant term (c/H_{D})P plus a sinusoidal term.
Since (∂X/∂t) = cA another integration with respect to time gives
The expression (c/ν) is in the nature of a wavelength, say λ. Then the solution is
The maximum value for λ is 3×10^{8}/1.5526×10^{21}=1.932×10^{-13} meters, or about 200 fermi.
For a one dimensional case here is the graph of X versus time.
(To be continued.)
This jittery motion is similar to what is implied for the solution to the simple cases, such as a harmonic oscillator, when the wave function is interpreted in terms of the time-spent probability density function.
In this graph the nearly flat portion represent what are called allowable quantum states; the steap portions of the trajectory represent what the Copenhagen Interpretation calls quantum jumps. They are not instantaneous transition between allowed quantum states but instead relatively rapid transitions.
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