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Dirac Equation VIII Zitterbewegung Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: April 24, 2017) Zitterbewegung is a hypothetical 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 an angular frequency of 2mc2 1.55268 x 1021 rad/s A reexamination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung. https://en.wikipedia.org/wiki/Zitterbewegung 1. Standard velocity (c) of a free particle The operator for velocity in the x direction can be computed from the commutator with the Hamiltonian. Using the Heisenberg’s equation of motion we have i x [H, x] i [c(α p) mc2 , x] i [c x px , x] c x Similarly, we have y c y , z c z or vi ci Using the Mathematica, we solve the eigenvalue for the velocity for vx c x For 1 c 1 0 1 0 1 1 u , u 1 2 0 2 2 1 1 0 For 2 c 1 0 1 0 1 1 u , u 1 2 0 2 2 1 1 0 In the Dirac theory, the velocity is c , while in the non-relativistic Pauli theory the velocity operator is 1 v p m Such a contradiction may be a good reason to suspect that the whole theory is up the creek. However, Foldy and Wouthuysen come to rescue our rescue again. The essential point is that these apparently contradictory operators do not actually represent quite the same observable. ((Note)) Sakurai’s comment (Advanced Quantum Mechanics) The plane-wave solutions which are eigenfunctions of p are not the eigenfunction of cα . Since cα fails to commute with the Hamiltonian, no energy eigenfunctions are expected to be simultaneous eigenfunction of cα . In the second quantization, the quantum field operator (r) can be expressed by the superposition of positive and negative energy plane wave solutions. c2 c2 p cp α dr (r) (r) p k 2 2 2 2 2 2 ER c m c p m c p which is time independent, represents the group velocity of the wave packet made of exclusively of positive- (negative-) energy plane-wave components. 2. Classical relativistic velocity of a free particle (FW component) We continue to use the Heisenberg’s equation of motion for the acceleration operator such that i vi [H,vi ] i [H,ci ] ic {[H,i ] 2i H} We note that 2 [H,i ] [cp j j mc ,i ] j 2 cp j{i , j} mc {i} j 2cp ji, j j 2cpi where 2 {i , j } 2ij I4 , {i ,} 0 , I4 where i = x, y, and z, the curly bracket denotes an anti-commutator, {i , j} i j ji Thus we get ic vi (2cpi 2i H ) ic 1 2H (i cpi ) H Here we introduce a new variable, 1 cp i i H i Since vi ci ic vi ci ci 2Hi Thus we have the first-order differential equation for i as i i 2Hi . The solution for this equation is 2i i (t) i (0)exp( Ht) or dx 1 i c c c2 p dt i i H i 2i 1 2 ci (0)exp( Ht) c pi H or 1 c 2i x (t) x (0) c2 p t (0)exp( Ht) i i H i 2iH i 1 2 c 1 2i xi (0) c pit [i (0) cpi ]exp( Ht) H 2iH H where 1 (0) (0) cp i i H i where xi (t) is the position operator at time t. The x-component of the velocity has two parts. The second term is the FW term and is associated with the average motion of the particle (classical relativistic formula). The first term is the Zitterbewegung which oscillates extremely rapidly. 3. Numerical calculation of zitterbewegung 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. 2 E mc2 E mc Radiation Radiation E 0 E 0 E mc2 E mc2 Fig. Dirac sea. The energy gap is E 2mc2 . The zitterbewegung is related to the transition between the upper level with the positive energy mc2 and the lower level with the negative energy mc2 . The angular frequency of zitterbewegung 2mc2 1.55268 x 1021 rad/s where mc2 =0.510997 MeV. The Compton wavelength is defined by 2 = 2.4263102367 x 10-10 cm =0.024263 Å C mc Note that the amplitude of the oscillation of the zitterbewegung 1 = 1.9308 x 10-11 cm 2mc c 4 3. Foldy–Wouthuysen term of the velocity (classical relativistic velocity) We note that the Heisenberg’s equation of motion is given by d i Aˆ [Hˆ , Aˆ] (1) dt where Hˆ c( p) mc2 So we have d i rˆ [Hˆ ,rˆ] cα (2) dt for r (velocity equal to the speed of light, c). We note that rˆFW for the position vector in the FW representation is defined by ˆ ˆ rˆFW UrˆU 2 (3) icα c r 2 2 {ic (α p) p ER (Σ p)]} 2ER 2ER (ER mc ) Here we introduce a new operator defined by i cα c2 ˆ ˆ ˆ ˆ R U rU r 2 2 {ic (α p) p ER (Σ p)]} (4) 2ER 2ER (ER mc ) ˆ ˆ Corresponding to R , the operator RFW is defined by ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ RFW URU UU rˆUU rˆ We also note that ˆ ˆ ˆ ˆ ˆ rˆFW UrU , R U rU ˆ ˆ ˆ ˆ ˆ ˆ U rˆFWU U UrˆU U rˆ Now we consider the Heisenberg’s equation of motion for Rˆ : dRˆ i [Hˆ , Rˆ ] dt or dRˆ dRˆ i i Uˆ Uˆ FW Uˆ[Hˆ , Rˆ ]Uˆ [UˆHˆUˆ ,UˆRˆUˆ ] dt dt or ˆ dRFW i ˆ ˆ [H FW , RFW ] dt i ˆ ˆ [H FW , RFW ] i [rˆ, ER ] c2 p ER or dRˆ c2 p FW dt ER where ˆ ˆ H FW ER , RFW rˆ ˆ Since RFW rˆ , we have drˆ c2 p dt ER This leads to 2 2 ˆ drˆFW drˆ c p c p H Uˆ Uˆ UˆUˆ dt dt ER ER ER dRˆ drˆ c2 p c2 p Hˆ Uˆ Uˆ Uˆ Uˆ dt dt ER ER ER Note that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ U ERU U H FWU U UHU U H or ˆ H Uˆ Uˆ ER ˆ Now the term H / ER has eigenvalue +1 for particle wavefunctions and -1 for antiparticle ˆ 2 wavefunctions. So, dR / dt drˆFW / dt is just c p/ ER for particles, which is the classical relativistic velocity. We also note that drˆ / dt cα (standard velocity) 4. Summary The standard velocity operator cα gives the electron speed as equal to the speed of light, while drˆFW / dt leads to a velocity that has a sensible non-relativistic limit. This means that the motion of the electron can be divided into two parts. First is the average velocity. Secondary there is very rapid oscillatory motion (Zitterbewegung) ____________________________________________________________________________ REFERENCES J.J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967). A. Wachter, Relativistic Quantum mechanics (Springer, 2011). A. Messiah, Quantum mechanics vol.I and II (North-Holland, 1961). W. Greiner, Relativistic Quantum Mechanics, Wave Equations (Springer, 1987). P. Strange, Relativistic Quantum mechanics (Cambridge, 1998). .
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