A Review on Dirac Jordan Transformation Theory

Transcription

1 Avaiabe onine at Avances in Appie Science Research, 01, 3 (4): A Review on Dirac Joran Transformation Theory F. Ashrafi, S.A. Babaneja, A. Moanoo Juybari, M. Jafari Matehkoaee ISSN: CODEN (USA): AASRFC Facuty of Sciences, Payame Noor University, Tehran, Iran ABSTRACT In this review we wi try to perform a comprehensive summery of Dirac Joran transformation theory. This theory inicates that, any quantum state of a partice wi etermine by a vector of state in abstract space, D. There are very few authors who have iscusse this theory in abstract an pure form. For this reason, the aim of this paper is to iustrate the abiity of this theory by ifferent eperiments, an to show its capabiity in ifferent consequences. Then, we wi try to show aso, the specific istinct between this metho an other methos. Keywors: Dirac Joran transformation theory, quantum state, vector of state, abstract space. INTRODUCTION Rarey, the authors of quantum mechanics books have iscusse Dirac Joran transformation theory [1] in abstract an pure form []. Mosty, in the topics of mathematica toos, quantum mechanics assimiates in a manner with matrices theory that an its operabiity an abiity ifferentiates as a pure theory is ifficut. The subject of this stuy is that to show the abiity of this theory in ifferent iscussions an particuar ifferencies of its soution methos with other theories. Principay, appie mathematics in Dirac Joran transformation theory is particuar an iffers with the mathematics present in theorems an reations of wave an matrices theories [3]. Encountering with wave an matrices theories maybe impies, at east, appie mathematics in these theories gives certain reation between them, but there is not the case of Dirac Joran transformation theory [4]. Quantum state of a partice in a given time, in Schröinger s wave theory, was efine by wave function ψ ( r ). Probabiistic interpretation of this wave function requires that its square cou be integrate, an this eas to stuy Hibert, H space [5 an 6]. By the viewpoint of this theory, a microscopic system may be efine by a wave ike equation [7]. Base on eementary suggestions of Schröinger, these waves may be eiste as Mawe waves. Later, it was carifie that, supposing these waves as rea ones is erroneous. In Dirac Joran's theory each quantum state of a partice, may be characterize by a state vector epening to an abstract space D (for respect of Dirac) which was name aso, state space of a partice [8]. D space is a sub- space of Hibert H space. Any eement or vector of D space, is name Ket vector or simpy, Ket an has show by, an for istinct with other Kets, a symbo is pace insie an was shown as ψ. D space of the states of a partice is efine by correating a Ket vector, ψ, of D to any function which may be integrate its square: 474

2 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): ψ (r) H ψ D (1) The movement of quantum partice is comparabe in some way to rotation of this vector. Seecting basic vectors which are forme in this space, there is a quantum efinition. It may be simpy going from one efinition to another an in fact, Dirac Joran's theory eas to perform this, an hence it was name transformation theory..1. Scaar prouct METHODS AND DEFINITIONS To each pair of Kets, ψ an cou be epene a compe conjugate which is a scaar prouct an has the foowing properties: ψ = (r) ψ (r)v () ψ = ψ (3) λψ λψ = λ1 ψ1 + λ ψ (4) λ + λ ψ = λ ψ + λ ψ (5) It is perfecty obvious that scaar prouct is inear with respect to ψ function, an is anti- inear with respect to function... D space Any eement or vector in D space, is name Bra vector or simpy, Bra an is shown as. Then, ψ symbo shows the position of both state vectors from two ifferent sub-spaces, or etermines the position of this two subspaces with respect to each other, so: ψ D D (6).3. There is a Bra reate with a Ket Presence of a scaar prouct wi enabe to show that an eement of D, i.e. a Bra, is reate to each Ket, Bra Ket reation is anti- inear because: ( ) λ + λ ψ = λ ψ + λ ψ = λ + λ ψ (7) λ λ + is the Bra epening to Ket, thus, Ket Bra reation is anti- inear. 1 1 λ λ D. If λ is a compe number, an ψ is a Ket, then λ ψ aso wi be a Ket an so: λψ = λ ψ (8) Aso, λψ efine a Bra epening to λψ Ket, because the reation between a Bra an a Ket is anti- inear, an this gives: = (9) λψ λ ψ 475

3 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): There is not a Ket reate with a Bra However, a Bra is correspon to a Ket, but by seecting eampes from Hibert space it is possibe to show that we can fin the Bras which aren t correspon to Kets. Wave function of free partice correspons with a Bra, but it is not correspon with any Ket. Consiering a fat wave which is efine by a momentum p o in the range of as () function ten to zero outsie of this range, so: 1 = πh ip o () e / h Reate Ket with (χ) is shown as, so that: () H D (11) (10) Then, it may be written as: 1 = πh πh ipo/ h ipo/ h e e (1) Which wi be ivergent with, thus in this case, any etreme in Hibert space is not efine. So that: im () H an im D (13) Now, consiering for an seecting again Ket from D space, 1 ψ = e πh ipo/ h ψ () (14) ψ D an thus we have: Right term in above equation, in imit, wi be Fourier transformation of ψ() for p = p o. im ψ = ( p ) (15) o Hence, when, ten towar a perfecty etermine Bra, im = D (16) p= po p= p o, so: RESULTS AND DISCUSSION 3.1. Representation of inverse operator of momentum in D space Consiering the effort performe for efine time as an operator in quantum mechanics, inverse operator of momentum has an important roe. Thus, its representation other than Fourier transformation may be effective. ip / Firsty, evoution operator cou be efine as h, then we have: T( ε ) = e ε h p ε / h h 1 iε h 1 Tε = e = 1+ p + o( ε ) + = + ε + o( ε ) + i p i p L L (17) h i p This resuts in: = Tε ε o( ε ) p L (18) h 476

4 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): Then: p ψ ε ε ψ ε ψ L h = T o( ) With ε 0 an ε we have: i i p ψ = h ψ = h ψ = h ψ (19) (0) Therefore: p ψ = h an, p = h ψ (1) () 3.. Dimensions of quantum space Assuming that ˆ operator is an eigenstate non zero with eigenvaue, so: = (3) The spectrum of ˆ operator may be stuie. Consiering property of evoution operator we have: [ ] T( λ) = + λ,,t( λ) = λt (4) Where T is evoution operator of transformation, an λ is a true constant. Then: ( λ ) T = + T (5) T is an eigenstate non zero, an + λ is an eigenvaue for. Thus, by starting from an eigenstate for, by appying T( λ ), by any true vaue, it may be estabishing another eigenstate for (in fact, λ may have any true vaue). Therefore, spectrum is a continuous one which is forme of a possibe vaues over true ais. This shows that in a D space with imite imension N, any an p observabes are not present which their repacer is iћ. Because, any time, by appying transformation operator, without any imitation λ wi be accrue. Evienty, eigenvaue for cou not be simutaneousy smaer than or equa to space imension an in same time is infinity. So, space imension, necessariy, must be infinity Two istinct Kets in D space Comparing two Kets an transformation theory, we can write: ψ = ψ ( ) (6) ( ) ψ = ψ = ψ ( ) (7) As we have: δ ( ) = ( ) an δ ( ) in D space, is usefu. Consiering the efinition of wave function in = + (8) 477

5 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): Then can write: = δ ( ) an ψ ( ) δ ( ) ψ ( ) = + (9) So that, the wave functions correspone to these both Kets are competey ifferent. Now we efine figure operator such as = an affect it on these both Kets (assuming that is eigenstate of ): = = k (30) ( ) ( ) = = = k (31) Where, k is a constant. It is evient that vector is in the same irection with basis Ket an vector is in the opposite irection Two mathematica methos in D space 1. Accoring to efinition of wave function in this theory ψ ψ ( ) = ( ) ( ) ( ) ( ) ψ = ψ (3) (33) Above integra may be sove by part to part integrating, as foowing: u = ( ) u = ( ) (34) = we have: v = ψ ( ) v = ψ ( ) (35) This gives: [ ( ) ψ + ( )] ψ ( ) ( ) (36) Therefore: ( ) ( ) ( ) ( ) ψ = ψ (37) ( ) = ( ) (38) = (39) Assuming that is a basic Ket, then we have: 478

6 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): = (40). Supposing a is eigenstate of  ( is an arbitrary operator), an this gives:  a = a a (41) An: a Aˆ a a = (4) Mutipying reation (40) by a, we obtain:  a a = a a a (43) Thus, we can concue: Â, a a = 0 (44) An this gives: f (A), a a = 0 (45) Consiering a a = a, then: f (A) = f (A) = f (A) (46) a a a a This equaity pays an important roe in perturbation theory in quantum mechanics. CONCLUSION In this paper the review of Dirac- Joran transformation theory by ifferent methos, an we have performe a comprehensive summery of this theory. We approach to show that any quantum state of a partice wi etermine by a vector of state in abstract D space. This stuy shows that ony by change in wave an matri theory symboizing, we obtain a new theorem which consists of a new epression (anguage) to epose quantum mechanics. Because not ony the pervious concepts may be iscusse, but, emonstration the concepts as space imension an ifferentiation between states, may be carry out stronger. Dirac- Joran's theory, at our point of view, is a strong epression, an this review may be a starting point for profoun stuies generaizing this theory. REFERENCES [1] Dirac, P.A.M. (January 197)., Proceeings of the Roya Society of Lonon, (011), A 113 (765), pp. 61. [] P.A.M. Dirac, The Principe of Quantum Mechanics, Carenon Press, Ofor, (1981). [3] Duncan A., Janssen M., Preprint submitte to Esevier Science, (1 November 009). [4]. Oppenheimer J. R., Physica review, (198), Vo. 13, pp. 66. [5] Nichoas Young, An introuction to Hibert space, Cambrige University Press, (1988). [6] Joachim Weimann, Linear operators in Hibert spaces, Grauate Tets in Mathematics, 68, Berin, New York: Springer-Verag, (1980). [7] Euar Prugovečki Euar, (1981), Quantum mechanics in Hibert space (n e.), Dover Books on Physics, (pubishe 006). 479

7 F. Ashrafi et a Av. App. Sci. Res., 01, 3(4): [8 J. J. Sakurai, Moern Quantum Mechanics, eite by San Fu Tuan, Benjamin/Cummings, Meno Park, CA, (1985). 480