An Introduction to the Study of Wave Mechanics by Louis De Broglie

By Louis De Broglie

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104) for n = (0, 0, −1). 103). 103) exhausts all the fields that allow such a reduction. 103) is not covariant. However, one can easily find its covariant generalization. 105) where ???????? is a constant isotropic four-vector, ???????? = (1, n), n2 = 1, ????2 = 0. 106) which is easy to verify. 105) means that ???????? is an eigenvector of the matrix ???????????? with the eigenvalue equal to (nE). e. 103). 103) in the following box form [32]: 1 + (n????) 1 − (n????) ????1 (????) + ????−1 (????)] ???? , 2 2 ???? + (????F)(????n) ˆ. 107) Here ???? is a normalizing factor, ????̂ ???? are components of the kinetic momentum operator, ???????? (????), ???? = ±1 are some functions of ????, and ???? is an arbitrary constant spinor, henceforth normalized as ????+ ???? = 1.

As a matter of fact the author of [307] used an operator that was a component of the spin pseudovector. In Ref. [78] the spin pseudovector was introduced, probably for the first time. An interesting approach to finding spin operators was suggested in Ref. [306] and applied, specifically, to reobtain the operator found in Ref. [78]. A noncovariant spin three-vector was constructed in Ref. [315], and a tensor spin operator was introduced in Refs. [79, 308]. These operators were studied in every detail in [96, 106, 148, 186, 205, 287, 305, 309, 320, 342].

133) in the following box form: ???? (????n) − ???? − ???? ????=( ) . 107). 107) at all. 32 | 2 Relativistic equations of motion Consider next the tensor spin operator studied in [79, 308] and [127, 297, 319, 320]. 137) and ???????????????? = ???????????? − ???????????? D̂ = ???? (???????? ????̂ ???? − ???????? ????̂ ???? ) − ???????????? ???????? ????????̂ = ????(−???????? , ???????? ), ˆ , ???????? = ????3 [???? × P] ???????? = ????3 ????????̂ 0 − ????????2 Pˆ . 138) For solutions ???? of the Dirac equation it obviously holds that ???????????? ???? = ???????????????? ????, and hence it makes no difference which of the two operators is used.

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