Integral-free Wigner functions

Transcription

1 Integral-free Wigner functions A. Teğmen Physics Department, Ankara University, 0600 Ankara, TURKEY Abstract arxiv:math-ph/070208v 6 Feb 2007 Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type ψq = e aq2 φq, the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples. PACS: w, Sq, d Introduction In a study of quantum corrections to classical statistical mechanics, Wigner constructed the function 2π ψ q 2 y ψq + 2 y e iyp dy. that now called by his name []. The goal was to replace the quantum mechanical wave function ψq with a probability distribution in phase space that providing a particlelike description of the underlying wave propagation. Hence the Wigner functionwf exhibits a representation of quantum states in phase space. Although in quantum mechanics due to the uncertainty principle, a certain point in phase space does not make sense, projection of the WF onto the q and p planes generates the relevant marginal distributions associated with that quantum state, where possible negative values correspond to non-classical situations [2]. WF has been much studied theoretically and experimentally since its introduction, not only in the context of quantum mechanics, but also in various branches of physics and related sciences. The WF given by can be obtained easily if the integral allow to get an answer analytically. In most cases, despite the existence of wave function ψq, the integral becomes quite cumbersome and sometimes it is impossible to handle it. The aim of the present paper is to present an alternative definition of the WF for the wave functions of a certain type, that is a derivational approach rather than the original integral based one. This new definition gives, at least partly, calculational tricks to overcome the difficulties mentioned above.

2 The approach considers a class of wave functions given in the form ψq = e aq2 φq a 0, 2 where a is a positive constant, and is based on the series expansion of φq in term of Hermite polynomials. In order to establish the convergence of the series we need the following theorem [3]: If φq is a piecewise smooth real function in every finite subinterval [ b,b] and if the integral is finite, then the series φq = e q2 φ 2 qdq 3 C n H n q, < q <, 4 n=0 where the coefficients C n can be computed by the orthogonality properties of the Hermite polynomials, converges to φq at every continuity point of φq. It is also assumed that in the case of complex φq the consequent of the theorem is still valid i.e., φq admits a uniformly convergent expansion on the basis of Hermite polynomials. With these assumptions it is possible to replace the integral by the series and finally to define the WF as the following φ q i 2πa e 2aq2 2 p φ q + i 2 p e p2 /2a 2, 5 which is the main result of this paper. Note that the assumptions ensure that φq is smooth enough tobeevaluatedover thedifferential operators. Itiseasytoshowthattheoperatorsφandφ in5are commutative andtheirproductisreal. Ausefulaspectofthisdefinitionisthatifφqcontainsafinite polynomial alone that may be an orthogonal polynomial such as Hermite or associated Laguerre polynomial, then obtaining the WFs for the individual states requires only simple differentiations without needing integral tables or computers. 2 Derivation With the choice of 2 the WF amounts to For our purpose, the Hermite polynomials takes the form dy φ q 2 y φq + 2 y e 2 2 ay2 iyp dy. 6 H n q ± 2 y = r n! q ± 2 yn 2r. 7 By the substitution of 4 and 7 into 6 we get r [m/2] n! s m! q 2 yn 2r q + 2 ym 2s e 2 2 ay2 iyp dy. 8 2

3 If one uses the Binomial expansion for the terms in the integral in 8, the WF takes the form n 2r α=0 n 2r α q n 2r α 2 m 2s α r [m/2] n! β=0 m 2s β s m! q m 2s β 2 β y α+β e 2 2 ay2 iyp dy, 9 } {{ } χp where the integral χp can be adopted to the integral [3] π t n e t2 +2itx dt = H i n 2 n e x2 n x. 0 Thus χp is straightforward; χp = i k π k+ 2 2 k e p2 /2a 2 H k p/ 2a, a where k = α+β. If we use the Rodrigues representation of the Hermite polynomials we get χp as Thus the WF yields 2πa e 2aq2 n 2r α=0 m 2s β=0 n 2r α m 2s β H k u = k e u2 k u e u2, 2 2π/a χp = i p α+β e p2 /2a 2. 3 q n 2r α 2 α i p α r [m/2] n! s m! q m 2s β 2 β i p β e p2 /2a 2. 4 The last two sums in 4 stand for q i 2 p n 2r and q + i 2 p m 2s respectively. With this arrangement it is obtained that 2πa e 2aq2 m=0 [m/2] C m Cn n=0 r n! r!n 2r! [2q i 2 p] n 2r s m! s!m 2s! [2q + i 2 p] m 2s p2 /2a 2 e. 5 Thus, with the help of 4 and 7, the Wigner function can finally be compacted as in the equation 5. 3

4 3 Applications 3. Gaussian wave packet The wave function for a system represented initially by a Gaussian wave packet is given as ψq = Cexp [ q q ] 0 2 e ip0q/, 6 4 q 2 where C = /[2π q 2 ] /4, q is the width of the packet centered at q 0 and p 0 is its average momentum. According to 2 [ ] φq = Cexp q2 0 e q 0q/[2 q 2] e ip 0q/ 7 4 q 2 and a = /[4 q 2 ]. Thus φ q i 2 p φ q + i [ ] 2 p = C 2 exp q2 0 e q 0q/ q 2 e p 0 p, 8 2 q 2 which is obviously a translation operator which converts p to p p 0. Therefore 5 and the fact that q p = /2 yield the WF as [ π exp q q ] 0 2 exp [ p p ] q 2 2 p 2 that confirms the correct result. 3.2 Harmonic oscillator The wave function corresponding to the Hamiltonian Ĥ = ˆp2 /2+ ˆq 2 /2 is given by where C n = [/π ] /4 / 2 n n!. By virtue of the equation [4] ψ n q = C n e q2 /2 H n q/, 20 H n [ q i 2 p ] H n [ q + i 2 p ] e p2 / = n 2 n n! L n [2q 2 +2p 2 / ] e p2 / 2 which is obtained by iteration, 5 gives the well known WF W n q,p = n π e 2H/ L n 4H/, 22 where H = p 2 /2+q 2 /2 and L n denotes the Laguerre polynomial of order n. 4

5 3.3 Singular oscillator The system is described by the Hamiltonian Ĥ = ˆp2 /2+ ˆq 2 /2+g 2 /ˆq 2 with g is a real constant. The normalized eigenfunctions of the Hamiltonian are ψ n q = C n q α L α /2 n q 2 / e q2 /2, 23 wherecn 2 = n!/[γn+α+/2 α+ ], ΓistheGammafunction, L α+/2 n is the associated Laguerre polynomial and α = /2+/4+2g 2 /2 [5]. The operator equation version of the WF is straightforward with the help of the argument presented up to now; α W n q,p = Cne 2 q2 / q p 4 L α /2 n [ q i p 2 2 ] L α /2 n [ q + i ] 2 p e p2 / For an arbitrary α and n, integral tables and the approach discussed here fail to get an exact WF of this system. Especially for the fractional values of α, even determining the ground state carries big difficulty since the action of the operator q p /4α on exp p 2 / is unknown. At least, I am not aware of it. But for some special values of α and n, WFs for this system can be obtained explicitly [4]. As a limit of the applicability of the method, consider the one dimensional anyon system [6], where the wave function ψ n q e q2 /2 q /2 H n q 25 corresponds to the Hamiltonian Ĥ = ˆp2 /2 /ˆq /ˆq 2. Obviously the matter arises from the term q /2 since the smoothness condition of φq is violated. Acknowledgments The author wishes to express his appreciations to T. Altanhan and B. S. Kandemir for their helpful discussions. This work was supported in part by the Scientific and Technical Research Council of Turkey TÜBİTAK. References [] Wigner E P., Phys. Rev [2] Hillery M., O Connel R F., Scully M O. and Wigner E P., Phys. Rep [3] Lebedev N N., Special functions and their applications, Dover Publications 972. [4] Tegmen A., Altanhan T. and Kandemir B S., Eur Phys J D [5] Perelomov A., Generalized coherent states and their applications, Springer, Berlin 986. [6] Hakobyan Y. and Ter-Antonyan V., Phys. Atom. Nucl