An introduction to the study of wave mechanics by Louis De Broglie PDF
By Louis De Broglie
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Ta it, A complete treatment of Grassmann algebra is given in the treatise by Berezin (19&5), A Grassmann algebra fs defined by a set of generators, which we denote by (&),ar = 1,. ,m. These generatws anticommute: .. so that, in part;icutar: e:=o. 26 SECONE) QUANTIZAT$OM AND COHERENT STATES The basis of the Grassmann algebra is made of at! distinct products of the generators. a,. einct bads elements are produced by the two poss5bifities of 'rncludlng a generator O or 1 t h e s for each of tz generatars.
F(e) and this Is the form we will obtain for the coherent state representation of a wave function. Sfmffarly, the eoherent state representation of an operator in the Grassmann algebra will be a funetfon of axtd and must have the form c* As for ordinaq complex functSons, a ddvattve can be detind for Grassmann variaMe functions. It is defined to be Edentkaal to the conptex derivative, except that in order for the derivative operator to act an 6, the mriable ( has to be anticommuted through until it is adjacent to For instance: &.
Since the basis (a)is orthononnal, the scalar product (nag is equal t o 6nalnb, 6napnbp which leads to: ... A cruciaf property of the coherent states is their overcompleteness in the Fock space. that is, the fact that any vector of the Fock space can be expanded in terms of coherent states. This is expressed by the closure relation where 1is the unit operator i n the Fock space, the measure is given by: and the integration extends over aft values of Re& and h#,. 2, one may verify Eq. 123) straightforwardly by integrating the left hand side t o obtain the familiar completeness refation Eq.
An introduction to the study of wave mechanics by Louis De Broglie