On 3D and 1D Weyl particles in a 1D box
- PDF / 428,388 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 17 Downloads / 195 Views
On 3D and 1D Weyl particles in a 1D box Salvatore De Vincenzoa Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, Caracas 1041-A, Venezuela Received: 23 February 2020 / Accepted: 29 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.
1 Introduction In 1928, in the first edition of his book in German [1], and in 1929, in a couple of articles [2,3], Weyl proposed — among other important things — two two-component wave equations for the description of free massless fermions in (3+1) dimensions [4,5] (an English translation of Ref. [3] can be seen in Ref. [6]). In 1957, Lee and Yang chose to assign one of these two equations specifically to the neutrino [7], but in 1958, Feynman and Gell-Mann showed that it was actually correct to assign the other equation to this particle [8]. Because there is now experimental evidence that a neutrino has a very small rest mass, the Weyl equations only approximately describe the behavior of this particle and its antiparticle. In passing, the (free) Weyl equations admit the standard minimal substitution and therefore admit an external electromagnetic four-potential; thus, these equations could also approximately describe the behavior of charged light fermions. In general, it can be said that the Weyl equations in (3+1) dimensions describe three-dimensional Weyl particles (i.e., 3D Weyl particles), and in (1+1) dimensions, they describe one-dimensional Weyl particles (i.e., 1D Weyl particles). The Weyl equations in (3+1) and (1+1) dimensions are more easily constructed from the respective Dirac equation (in its respective Weyl representation). A particularly nice
I would like to dedicate this paper to the memory of my beloved father Carmine De Vincenzo Di Fresca, who passed away unexpectedly on March 16, 2018. That day something inside of me also died. a e-mail: [email protected] (corresponding author)
0123456789().: V,-vol
123
806
Page 2 of 18
Eur. Phys. J. Plus
(2020) 135:806
derivation of th
Data Loading...
