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Zitterbewegung, the trembling
motion of a moving electron

Dirac's Equation of an Electron

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

ih(∂ψ/∂t) = HDψ
where
HD = A·P

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

P = (mc², p1c, p2c, p3c)

where m is the mass of the electron and pj is the momentum along the j-th axis. The operator for P is found by substituting −ih(∂/∂xj for pj. Thus

P^ = (mc², −ih(∂/∂x1)c, −ih(∂/∂x2)c, −ih(∂/∂x3)c)

The function HD is known as the Dirac Hamiltonian.

The Heisenberg Picture (Representation)
in Quantum Theory

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

h(∂Q/∂t) = i[H, Q]

where the bracket symbol is defined as

[C, B] = CB − BC

Let X be the particle position vector (x1, x2, x3,). Now consider

[HD, xj]

Since (∂xj/∂xk)=δjk when AP^ operates on xj the only nonzero term is

hcαj

On the other hand since αj is a constant xjAP^ has no nonzero terms. Thus

[HD, xj] = hcαj
and hence
h(∂xj/∂t) = hcαj

Combining all such terms gives

h(∂X/∂t) = hcA
or, equivalently
(∂X/∂t) = cA

Thus cA is in the nature of a velocity matrix.

The velocity operator is obtained in the Heisenberg picture by defining

A^ = exp(iHDt/h)Aexp(−iHDt/h)

The time-dependence of this velocity operator is given by

h(∂A^/∂t) = i[HD, A^]

Elsewhere it shown that

αjαk = −αksαj for j≠k
but
αjαj = I

where I is the appropriate identity matrix.

This means that

[HD, A^] = (2/h)(PA^HD)

Thus

h(∂A^/∂t) = 2i(PA^HD)

The above formula is confirmed by a mathermatical manipulation. The term [HD, A^] can be expressed as

[HD, A^] = (2/h){HDA^ + A^HD) − 2A^HD}

In Albert Messiah's Quantum Mechanics it is shown that

½(HDA^ + A^HD) = P

For a free electron the parameters P and HD are constants and the above equation can be integrated with respect to time to give

A(t) =W·exp(iνt) +(c/HD)P

where W is equal to (A(0) − (c/HD)P) and the frequency ν is 2HD/h. The minimum value of HD 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=1.5526×1021 per second.

So velocity is equal to a constant term (c/HD)P plus a sinusoidal term.

Since (∂X/∂t) = cA another integration with respect to time gives

X(t) = X(0) + (c²/HD)Pt
+ (ic/ν)W(exp(iνt) − 1)

The expression (c/ν) is in the nature of a wavelength, say λ. Then the solution is

X(t) = X(0) + (c²/HD)Pt + iλW(exp(iνt) − 1)

The maximum value for λ is 3×108/1.5526×1021=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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