Special functions and quantum mechanics in phase space: Airy functions

Transcription

1 PHYSICAL REVIEW A VOLUME 53, NUMBER 6 JUNE 996 Special functions and quantum mechanics in phase space: Airy functions Go. Torres-Vega, A. Zúñiga-Segundo, and J. D. Morales-Guzmán Departamento de Física, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Apartado Postal 4-74, 7 México, Distrito Federal, Mexico Received 3 October 995; revised manuscript received 4 February 996 We look for the solution to the eigenvalue problem, in a recently introduced phase-space representation, for the quantum particle in a linear potential. We find that the solution is not unique and that some of these functions correspond to the eigenfunctions in coordinate space and that others correspond to the classical limit. S PACS numbers: 3.65.w I. INTRODUCTION it 3 /3t 2 /2itqip Aip,q; eipqip/2 A recent phase-space representation of quantum mechanics that postulates wave functions dependent on both coordinate and momentum of a given system complies with the requirements for a quantum representation and allows us to analyze quantum evolution in phase space in a way similar to how it is done in coordinate, momentum, or abstract spaces has been introduced,2. In contrast to Wigner 3, Husimi 4, Bargaman 4, or other spaces, the phase-space representation of Ref. so far has shown to comply with the standard requirements for a nonrelativistic quantum-mechanical representation, without introducing additional quantities and without the complications that other formulations have. This representation is simple enough as to allow the analysis of the dynamics of quantum systems completely in phase space, formally as well as numerically, without recurring to coordinate representation. Now, a phase-space representation of quantum mechanics leads to the question of the existence of phase-space versions of the special functions normally used in the coordinate space. In a previous paper, we have found the phase-space eigenfunctions for the case of the harmonic oscillator and in this paper we look for the phase-space eigenfunctions for the quantum particle in a linear potential. In Sec. II we introduce a set of two-variable special functions with properties similar, and that can be reduced, to the usual Airy functions. These functions are the solution to a two-variable partial differential equation. In Sec. III a brief review of the recently introduced phase-space representation of nonrelativistic quantum mechanics is made. In Sec. IV the eigenvalue problem for the phase-space quantum particle in a linear potential is solved. We find that some solutions resemble the coordinate wave function and others resemble the classical limit. Finally, in Sec. V some concluding remarks are made. II. TWO-VARIABLE AIRY FUNCTION The two-variable Airy function is defined by 2 eipqip/2 t 2 /2 cos t3 3 tqip, where. Note that when, the above function, aside from a phase factor, becomes the usual one-variable Airy function. This function satisfies a second-order partial differential equation of particularly simple type. By differentiating under the sign of integration, we find that p i 2 q 2Aip,q; q i Aip,q; 2 p ie ipqip/2 e it3 /3t 2 /2itqip t t. At the extremities t the quantity in square brackets vanishes. Therefore the equation p i 2 q 2w q i w 2 2 p is satisfied by wai(p,q;). As we can see, the solution to this equation is not unique, but it is a family of solutions that depend on the parameter. In Figs. and 2 we show density plots of the square magnitude and phase of the two-variable Airy function Eq., for and.5, respectively. For the square magnitude, darker regions indicate that the function has a large value and in the white region the function is very small. For the phase plot, large negative values are indicated by white regions and the darkest regions indicate where the phase has the largest positive value. Other equations for this function are q i 2 p i p i 2 qaip,q; qipaip,q;, /96/536/37926/$ The American Physical Society

2 53 SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE FIG.. Density plots of the square magnitude and phase of the two-variable Airy function for. which looks like an eigenvalue equation with the eigenvalue qip, and FIG. 2. Density plots of the square magnitude and phase of the two-variable Airy function for.5. dt e it 3 /3t 2 /2itqip q Re Ai*p,q; p 2 i Aip,q; q e 2i/3 dt exp i 3 t3 2 e2i/3 t 2 itqe 2i/3 p Aip,q;2, 3 ie 2i/3 pe 2i/3, 5 which is a conservation equation for the density Ai(p,q;) 2. Some of the properties of these functions can be easily derived from the equalities obtained by integrating expiv 3 /3v 2 /2iv(qip) around the contours C i shown in Fig. 3. The needed results are dt e it 3 /3t 2 /2itqip e 2i/3 dt exp i 3 t3 2 e2i/3 t 2 itqe 2i/3 ie 2i/3 pe 2i/3, 4 FIG. 3. Contours used in the text.

3 3794 GO. TORRES-VEGA et al. 53 dt e t 3 /3t 2 /2tqip e i/6 dt exp i 3 t3 2 e2i/3 t 2 itqe 2i/3 ie 2i/3 pe 2i/3, 6 t 3 /3t 2 /2tqip e i/6 dt exp i 3 t3 2 e2i/3 t 2 itqe 2i/3 ie 2i/3 pe 2i/3. 7 A power-series expansion for Airy s function is given by Aip,q; 2 eipqip/2 n n! qip n t 3 /33t 2 / t n cos 2 5 t2 n Now, Eq. 2 is unaffected when (p,q) is replaced by (pe 2i/3,qe 2i/3 ); hence other solutions are obtained by making these replacements. However, the functions so obtained are not independent, but are related as Aip,q;e 2i/3 Aipe 2i/3,qe 2i/3 ;e 2i/3 e 2i/3 Aipe 2i/3,qe 2i/3 ;e 2i/3, as can be easily verified by combining Eqs Another solution, which we denote by Bi(p,q;), to Eq. 2 is defined by the combination Bip,q;e i/6 Aipe 2i/3,qe 2i/3 ;e 2i/3 e i/6 Aipe 2i/3,qe 2i/3 ;e 2i/3 eipqip/2 dt e t3 /3t 2 /2tqip e t2 /2 sin t3 3 tqip. 8 In Figs. 4 and 5 we show density plots of the square magnitude and phase of the second solution Eq. 8 for the cases and.5, respectively. These functions increase rapidly in the region q. As for the case of the functions Ai(p,q;), by replacing (p,q) with (pe 2i/3,qe 2i/3 )we obtain another two solutions. However, these solutions are not independent; in fact, FIG. 4. Density plots of the square magnitude and phase of the second kind two-variable Airy function Bi(p,q;), for. Bip,q;e 2i/3 Bipe 2i/3,qe 2i/3 ;e 2i/3 e 2i/3 Bipe 2i/3,qe 2i/3 ;e 2i/3. One last relationship between these functions is Aipe 2i/3,qe 2i/3 ;e 2i/3 2 ei/3 Aip,q;Bip,q;. III. QUANTUM-MECHANICAL PHASE SPACE In this paper, we make use of the recently introduced phase-space representation of non-relativistic quantum mechanics,2 in which the operators associated with the momentum Pˆ, coordinate Qˆ, and inverse coordinate Qˆ operators are given by Pˆ p i 2 q, Qˆ i eipq/2 Qˆ q i 2 p, p dpe ipq/2. The operators Pˆ p/2i/q and Qˆ q/2i/p do not commute with each other; in fact, Qˆ,Pˆ i and so far

4 53 SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE m p i 2 q 2V q i 2 p E E E or numerically, by diagonalizing the matrix elements of the Hamiltonian operator, in a given basis, or by propagating a nonstationary initial phase-space wave function and utilizing the standard time-dependent formalism, which requires the evaluation of the Fourier transforms lim T T T dt exp(it) t and E lim T T 2T dt e iet/ t. T One can recover the usual coordinate or momentum spaces from the phase-space representation. The wave function in coordinate space q can be recovered from the wave function in phase space by means of the projection q exp(ipq/2)dp and the momentum wave function is obtained from the wave function in phase space by means of the projection p exp(ipq/2)dq. Similarly, the phase-space matrix elements p,qôp,q of an operator Ô can be projected to coordinate or momentum spaces by using the equalities qôq 2 du dse iqs/ us,qôus,q FIG. 5. Density plots of the square magnitude and phase of the second kind two-variable Airy function Bi(p,q;), for.5. this formulation has shown to comply with the requirements for a quantum representation. The basis vectors are, where (p,q) denotes a point in phase space, and the projection () is the phase-space wave function, with its complex conjugate given by the projection *(). The quantity () 2 is, by definition, a non-negative quantity that can be considered to be the quantum probability density in phase space. The inner product between two phase-space functions () and () is defined in the usual way *()()d. Then, the phase-space Schrödinger equation is given by i t 2m p i 2 q 2V q i 2 p, where V(q/2i/p) is the potential function evaluated at q/2i/p. Within this representation one can analyze, formally as well as numerically, quantum dynamics entirely in phase space in the same way as it is done in coordinate or abstract spaces, without introducing additional quantities and complications into the theory. For instance, finding eigenvalues and eigenfunctions of the Hamiltonian operator can be done in a way similar to how it is done in coordinate representation: by analytically solving the eigenvalue problem and pôp 2 du dse ips/ p,usôp,us. The diagonal matrix element of the quantum probability conservation equation is t ˆ q 2m Pˆ ˆˆPˆ p n n V nm n l n V n l Qˆ l ˆQˆ nl Qˆ l ˆQˆ nl, where ˆ is the time-dependent density operator and we have assumed that the potential function can be written as V(q) nm V n q n for some integer M. Note that in the above equation, the term V is no longer there and it is combining the corresponding equations in coordinate t and momentum qˆ q q 2m qpˆ ˆqqˆPˆ q 9

5 GO. TORRES-VEGA et al FIG. 7. Classical stationary density F c E ~ G; a! 5NAi FIG. 6. Phase-space probability and flux densities for the quantum particle in a linear potential. Cases a 5.,,3. The solid line is the classical trajectory p 2 q5. ^ p u r u p & 5 t p F( n5m ` ( n5 n V 2n ( ^ p u Q 2l r Q 2nl2u p & l5 n2 Vn ( l5 ^ p u Q l r Q n2l2 u p & S DS p 2mK, \2 ~ 2mK\! /3 /3 q2 D G 2E ;ab, K ~! where N is the normalization factor. We have chosen the function Ai( p,q; a ), instead of Bi(p,q; a ), because it is the nondivergent solution. In Fig. 6, there are plots of the dimensionless probability density u Ai(p,q; a ) u 2, Eq. ~!, and of the corresponding Eqs. ~3! and ~9!# for a 5.,,3. As can be seen in this figure, for small value of a 5., the square magnitude u Ai(p,q; a ) u 2 resembles the usual quantum coordinate probability for this system in the q direction, whereas it is very broad in p. The solid line is the classical trajectory p 2 q5, and we can notice that, for this value of a, there is a lot of interference ~or tunneling between the two branches of the classical trajectory! where the probability is large. For a 5 ~see Fig. 6!, the coordinate q and momentum p have the same weight in u Ai(p,q; a ) u 2 and the probability begins to be centered around the classical trajectory G spaces, providing a better description of quantum dynamics. IV. QUANTUM PARTICLE IN A LINEAR POTENTIAL According to the preceding section, the phase-space, time-independent Schro dinger equation for a particle in a linear potential of strength K is F S p 2i\ 2m 2 q D S 2 K q i\ 2 p DG c E ~ G! 5E c E ~ G!. This equation can be put in the form of Eq. ~2! by making the replacements p p(2mk\) /3 and q (\ 2 /2mK) /3q 2E/K. Then the eigenfunction, with eigenvalue E, for the particle in a linear potential is given by FIG. 8. Wigner function equivalent to the probability densities in Fig. 6. The solid line is the classical trajectory p 2 q.

6 53 SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE p 2 q. The interference or tunneling has decreased substantially with respect to the case.. For the large value 3 see Fig. 6, the probability is definitely centered around the classical trajectory and the interference is gone, imitating classical behavior. In order to see how far from a classical stationary density is the quantum density for 3, in Fig. 7 there is a density plot of the classical stationary density exp(p 2 q) 2 /3. These plots are almost identical. Thus the functions introduced in this paper can be used as a stationary solution in a variety of situations ranging from the quantum coordinatelike small value of ) to the quantum phase-space-like equal weight for p and q, i.e., ) to a classical-like large value of ) density. In this case, the classical limit corresponds to a large value of the free parameter. V. CONCLUDING REMARKS We can make a comparison between the above results and other phase-space functions. For instance, in Fig. 8 we show a density plot of the Wigner function where Wp,q 22/3 2 Ai2 2/3 p 2 q, Aix e it 3 /3ixt dt is the usual Airy function, which is the solution to the ordinary differential equation d 2 dx 2 wxxwx. This is the equivalent to the calculation shown in Fig. 6. As is usual for Wigner functions, this density is not always positive and has peaks at many places aside from near the classical trajectory, with wild oscillatory behavior, hindering the identification of correlation between coordinate and momenta. There is a big difference between the functions introduced in Sec. II and Eq.. The Wigner function Eq. is just the usual Airy function, which is the solution to an ordinary differential equation the Schrödinger equation in coordinate space with argument 2 2/3 (p 2 q). In contrast, the functions of Sec. II are solutions to a two-variable partial differential equation: a Schrödinger equation in phase space. The functions used in this paper seem to be appropriate for a full quantum phase-space analysis, which includes a coordinatelike wave function as well as a classical-like wave packet, depending on the value of a free parameter. The introduction of orthogonal polynomials and special functions in phase space shows that it is possible to deal with quantum systems completely in phase space, in a fashion similar to how it is done in abstract, coordinate, or momentum spaces. This means that one can have quantum wave functions that depend on both coordinate and momentum variables without violating any of the quantum principles and allows us to use the methods of standard nonrelativistic quantum mechanics. All of these things can be done in a very simple manner without the complications found in other formulations. ACKNOWLEDGMENTS We would like to acknowledge financial support from CONACyT and SNI, Mexico. Go. Torres-Vega and John H. Frederick, J. Chem. Phys. 93, ; 98, ; Phys. Rev. Lett. 67, 26 99; Go. Torres-Vega, J. Chem. Phys. 98, ; 99, ; Go. Torres-Vega and J.D. Morales-Guzmán, ibid., Qian-Shu Li and Xu-Guang Hu, Phys. Scr. 5, E. Wigner, Phys. Rev. 4, ; M. Hillery, R.F. O Connell, M.O. Scully, and E.P. Wigner, Phys. Rep. 6, 2 984, and references therein. 4 K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, ; Coherent States, edited by J.R. Klauder and B.S. Skagerstum World Scientific, Singapore, 985, and references therein.