Dirac electron in orthogonal electric and magnetic fields

Abstract

The p rob lem of a Dirac e lec t ron in ex t e rna l c o n s t a n t electr ic and magne t i c fields o r thogona l to each o the r has recen t ly been discussed (1) and ga ined wide app l ica t ions in t he l i t e ra tu re . However , t he only avai lable solut ion as g iven in CC is b o t h incomple te and incorrect . In pr inciple , if we exclude the case E = H, i t is no t difficult to see t h a t e x a c t solut ions do exis t for a Dirac e lec t ron in crossed electr ic and magne t i c fields. One only has to observe these fac ts : 1) E x a c t solut ions for Di rae ' s equa t ion exis t for c o n s t a n t m a g n e t i c field (2). 2) E x a c t solut ions also exis t for c o n s t a n t electric field (3). 3) F o r E ~ H a n d E . H = 0, a Lo ren t z t r a n s f o r m a t i o n can a lways be made to m a k e the e lec t r ic field van i sh (4). 4) Fo r E > H and E . H = 0, a L o r e n t z t r a n s f o r m a t i o n can always be m a d e to make the magne t i c field van i sh (4). The res t r ic t ion E ¢ H has i ts origin in t he rea l iza t ion of t he Loren tz t r an s fo rma t ion . In fact , in CC the cond i t ion E ~ H is impl ic i t ly used in t he definit ion of fl in eq. (5) to make f l ~ c or to make y real so t h a t the ene rgy e igenvalues found in cq. ( l l ) are real and make sense. The case E ~ H , which is d ras t ica l ly d i s t inc t f r o m t h a t of E < H as shown below is o m i t t e d in CC. E v e n wi th the condi t ion E < H in mind , the solut ion given in CC canno t be correct for t he reason t h a t the m a t r i x S which is supposed to diagonalize A in the i r eq. (5) as c la imed does no t exis t a t all. This is because