Eigenvalue equation for momentum dispersion operator and properties of phase space wavefunctions

Transcription

1 Eigenvalue equation for momentum dispersion operator and properties of phase space wavefunctions Ravo Tokiniaina Ranaivoson 1, Raoelina Andriambololona 2, Hanitriarivo Rakotoson 3 raoelinasp@yahoo.fr 1 ;jacquelineraoelina@hotmail.com 1 ;raoelina.andriambololona@gmail. com 1 ; tokhiniaina@gmail.com 2,infotsara@gmail.com 3 Theoretical Physics Department Institut National des Sciences et Techniques Nucléaires (INSTN- Madagascar) BP Antananarivo Madagascar, instn@moov.mg Abstract: This paper is a continuation of our previous works concerning coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It deals with the properties of momentum dispersion operator, its representations, eigenvalue equation and its relations with the wavefunctions in the phase space representation. After recalling some results from our previous works, we perform a study concerning the phase space wavefunctions and consider some example of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any wavefunction in phase space representation is solution of this equation. Keywords: Quantum mechanics, eigenvalue equation, dispersion operator, phase space representation, wavefunction 1- Introduction The present work can be considered as part of a series of studies related to a phase space representation of quantum theory introduced in [1]. It is a direct continuation of our paper untitled coordinate, momentum and dispersion operators in phase space representation [2] and is related to [3]. Because of the uncertainty relation [4], the problem of considering phase space, which mix momentum with coordinate, in quantum theory is an interesting challenge. Many works related to this subject have been already performed. We may quote for instance [5-12]. Many of these works are based on the approach introduced by Wigner in [5]. In the reference [1], we have considered an approach using the current formulation of quantum mechanics based on linear operators theory and Hilbert space (see for instance [4]). Our present work [1] may be considered as extension in the framework of quantum theory of the results obtained in [14]. The phase space representation that we have defined is based on the introduction of quantum states, denoted,,,. These states are defined by the means values, and statistical dispersions (variance) and of coordinate and momentum being a positive integer number). We have the relations 1

2 ,,,,,,,,,,,,,,,,,,21,,,,,, and being respectively the momentum and coordinate operators. The wavefunctions corresponding to a state,,, respectively in coordinate and momentum representations are the harmonic Gaussian functions and their Fourier transforms defined in [1] and [14],,,,,, 2! 2,,,,,, 1 2,,, 2! is a Hermite polynomial of order. It has been established that a state,,, is an eigenstate of the momentum dispersion operator and coordinate dispersion operator. The explicit expressions of these operators are and the corresponding eigenvalues equations are,,,,,, 21,,,,,,,,, 21,,, 1.5 As the eigenvalues of the momentum and coordinate dispersion operators are proportional and as they have the same eigenstates, it is sufficient to consider only the momentum dispersion operator. In the reference [2], we have tackled the problem of finding the representations of coordinate, momentum and dispersion operators in the frameworks of the phase space representation. We 2

3 have established that they can be at the same time represented both with matrix and differential operators in the basis,,, defining this phase space representation. Our goal is to study the relation between the eigenvalue equation of momentum dispersion operator and the wavefunctions in this phase space representation. We show and verify explicitly, in particular, that these wavefunctions are as expected the eigenfuctions of the differential operator representation of. 2-Phase space representations of momentum dispersion operator Let us consider the momentum dispersion operator. It can be put in the form in which In the paper [2], we have established that in the phase space representation, we have for the operators and on one hand the matrix representations,,,,,, 1 2 1,,,,,, 2 1 which satisfy the commutation relation and on the other hand the differential operators representations:, From the relation (2.2) and (2.3a), we obtain for the matrix representation of the operator,,,,,, and from the relation (2.2) and (2.4), we obtain for the differential operator representation

4 Using the relation (2.1), (2.5), (2.6) and (1.1), we obtain respectively for the matrix and differential operator representations of the momentum dispersion operator,,,,,, We may remark that the expression of corresponds to the fact that the elements of the basis,,, defining the phase space representation are the eigenvectors of. 3-Phase space wavefunctions A phase space wavefunction Ψ,, of a particle is a phase space representation of a vector state i.e a component of the vector in the basis,,,. As established in our paper [1], we have explicitly the relations Ψ,,,,,,,,,,, 3.1 Ψ,,,,,Ψ,,,,, The functions and are the wavefunctions corresponding to the state respectively in coordinate and momentum representation. The functions,,, and,,, are the complex conjugate of the wavefunctions,,, and,,, corresponding to the state,,, respectively in coordinate and momentum representation. The expressions of and are given in (1.3a) and (1.3b). Three particular examples of phase space wavefunctions are for Ψ,,,,,,,, 3.3 for Ψ,,,,,,,, 3.4 with for,,, Ψ,,Φ,,, 3.5 Φ,,,,,,,,,,,,,,, 3.6 4

5 as we have for any state Ψ,,,,, 3.7 It follows in particular from the relation (3.5) that we have the relation,,,φ,,,,,, 3.8 Let us now consider any state. In the basis,,, we have and in the basis,,, Ψ,,,,, 3.9 Ψ,,,,, 3.10 By inserting the relation (3.8) in (3.10) and identifying with (3.9), we may deduce an interesting property which holds true for any phase space wavefunction Ψ,, Ψ,,Φ,,, Ψ,, 4-Differential equation satisfied by phase space wavefunctions 3.11 On one hand, from the first part of the relation (2.7) and the relation (3.9), it can be deduced that for any phase space wavefunctions Ψ,,,,, we have the relation,,, 21 Ψ,,21Ψ,, 4.1 and on the other hand, from the definition of differential operator representation, as given in our work [2], we have,,, Ψ,, 4.2 In which is the differential operator representation of the momentum dispersion operator given in the second part of the relation (2.7) It follows from the relations (4.1), (4.2) and (4.3) that any phase space wavefunction Ψ,, satisfies the differential equation 5

6 Ψ 21Ψ 4.4 This equation is also the eigenvalue equation for the differential operator representation of the momentum dispersion operator and according to it, any phase space wavefunctions Ψ is an eigenfunction of with the eigenvalue equal to 21. The above results can be also verified explicitly using the properties (3.11) of the phase space wavefuctions. In fact, according to this relation we have Ψ,,Φ,,, Ψ,, So, taking into account the expression (4.3) of, we have But according to the relations (3.6) Ψ,, Φ,,, Ψ,, 4.5 Φ,,,,,,,,,,,,,,, and taking into account the expression (4.3) of, we have Φ,,,,,,,,, 4.6 Using the explicit expression of given in (1.3a) and the expression of given in (4.3), we can establish after a long but straightforward calculation the relation,,,21,,, 4.7 This relation means that the function,,, is an eigenfunction of with the eigenvalue 21. Introducing the relation (4.7) into (4.6) and taking account of (4.5) and (4.4), we obtain, as expected, an explicit checking of (4.4). We may remark that according to (3.3), the relation (4.7) can also be considered as a particular case of (4.4). 5- Conclusion The present work has shown explicitly that any phase space wavefunction Ψ,, is an eigenfunction of the differential operator, which is a representation of the momentum dispersion operator in the phase space representation, with eigenvalue equal to 21 as expected. 6

7 The corresponding eigenvalue equation given in the relation (4.4) is a second order differential equation in the variables and. These results can be considered as a consequence of the fact that, as indicated by the relation (2.7), the dispersion operator can be at the same time represented either with a diagonal matrix or with a differential operator in the basis,,, defining the phase space representattion. References 1. Ravo Tokiniaina Ranaivoson, RaoelinaAndriambololona, Rakotoson Hanitriarivo, Roland Raboanary, "Study on a Phase Space Representation of Quantum Theory", arxiv: v3 [quant-ph], International Journal of Latest Research in Science and Technology Volume 2, Issue 2: pp26-35, Hanitriarivo Rakotoson, Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Raboanary Roland, "Coordinate, momentum and dispersion operators in phase space representation", arxiv: [quant-ph],international Journal of Latest Research in Science and Technology ISSN (Online): Volume 6, Issue 4: Page No. 8-13, July-August Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson, Damo Emile Randriamisy, "Dispersion Operator Algebra and Linear Canonical Transformation", arxiv: v2 [quant-ph], International Journal of Theoretical Physics,Volume 56, Issue 4, pp , Springer, April Raoelina Andriambololona, "Mécanique quantique", Collection LIRA, Institut National des Sciences et Techniques Nucléaires (INSTN- Madagascar), E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev 40, , H.J. Groenewold, "On the Principles of elementary quantum mechanics", Physica12, J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society 45, , G Torres-Vega, J.H. Frederick, "A quantum mechanical representation in phase space", J. Chern. Phys. 98 (4), H.-W. Lee, "Theory and application of the quantum phase-space distribution functions", Phys. Rep 259, Issue 3, , K. B Moller T. G Jorgensen, G. Torres-Vega, "On coherent-state representations of quantum mechanics: Wave mechanics in phase space". Journal of Chemical Physics, 106(17), DOI: / , A. Nassimi, "Quantum Mechanics in Phase Space",arXiv: [quant-ph], T.L Curtright,C.K. Zachos, Quantum Mechanics in Phase Space,arXiv: v2 [physics.hist-ph], D. K. Ferry, Phase-space functions: can they give a different view of quantum Mechanics, Journal of Computational Electronics, Volume 14, Issue 4, pp , December Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. "Time- Frequency analysis and harmonic Gaussian functions", Pure and Applied Mathematics Journal.Vol. 2, No. 2,2013, pp doi: /j.pamj