A New Twist on Relativistic Electron Vortices

Abstract

There are important differences between the nonrelativistic and relativistic description of electron beams. In the relativistic case the orbital angular momentum quantum number cannot be used to specify the wave functions and the structure of vortex lines in these two descriptions is completely different. We introduce analytic solutions of the Dirac equation in the form of exponential wave packets and we argue that they properly describe relativistic electron beams carrying angular momentum. DOI: 10.1103/PhysRevLett.118.114801 Introduction.-Recent advances in experiments with relativistic (100-300 keV) electron beams [1-9] carrying orbital angular momentum call for a mathematical description based on the Dirac equation. The generally used Schrödinger equation gives an inadequate description because the differences between the nonrelativistic and relativistic wave functions are essential. It is not only the problem of relativistic corrections, which for 300 keV electrons may amount to about 60%. A more important difference is in the use of the orbital angular momentum quantum number l in the description of electronic states. In the nonrelativistic case both the orbital angular momentum and the total angular momentum are separately conserved while in the relativistic case only the total angular momentum has this property. This has already been pointed out by Dirac who in his first paper on the theory of the electron wrote "This makes a difference between the present theory and the previous spinning electron theory, in which m 2 is constant." Directly related to the problem of orbital angular momentum is a different structure of vortex lines in the two cases.The nonrelativistic wave function in free space is simply a product of the coordinate part and the spin part; the orbital angular momentum and the spin are separately conserved. In the relativistic theory, even in free space, the spin is coupled to the orbital angular momentum and only the total angular momentum is conserved. As a consequence, there are no acceptable solutions of the Dirac equation that are eigenstates of the orbital angular momentum.The proof of this assertion starts with the Dirac equation iℏ∂ t Ψ ¼ HΨ and we assume that L z Ψ ¼ ℏlΨ. By multiplying both sides of the Dirac equation first by ℏl, then by L z , and subtracting the two equations one obtains ½L z ; HΨ ¼ 0. Obviously, also ½L z ; H 2 Ψ ¼ 0. Hence, ðp

‘Twisted’ electrons

“…As discussed in the last section of this work, analogues to these types of interactions caused by purely relativistic effects have also been recently put forward, thus introducing a wide range of exciting phenomena. Such relativistic analyses result in wavefunctions that have completely different structures than their nonrelativistic counterparts [23,[71][72][73][74] and hence, in some cases, affect the nature of their quantum dynamics. For instance, the presence of spinto-orbit coupling in relativistic electron wavepackets directly affects the description of its coiling probability density current.…”