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Special functions and quantum mechanics in phase space: Airy functions
Go. Torres-Vega, A. Zu´niga-Segundo, and J. D. Morales-Guzma´nDepartamento de Fı´sica, Centro de Investigacio´n y de Estudios Avanzados del Instituto Polite´cnico Nacional, Apartado Postal 14-740,
07000 Mexico,Distrito Federal, Mexico
~Received 30 October 1995; revised manuscript received 14 February 1996!
We look for the solution to the eigenvalue problem, in a recently introduced phase-space representation, forthe quantum particle in a linear potential. We find that the solution is not unique and that some of thesefunctions correspond to the eigenfunctions in coordinate space and that others correspond to the classical limit.@S1050-2947~96!05106-2#
PACS number~s!: 03.65.2w
I. INTRODUCTION
A recent phase-space representation of quantum mechan-ics that postulates wave functions dependent on both coordi-nate and momentum of a given system complies with therequirements for a quantum representation and allows us toanalyze quantum evolution in phase space in a way similar tohow it is done in coordinate, momentum, or abstract spaceshas been introduced@1,2#. In contrast to Wigner@3#, Husimi@4#, Bargaman@4#, or other spaces, the phase-space represen-tation of Ref.@1# so far has shown to comply with the stan-dard requirements for a nonrelativistic quantum-mechanicalrepresentation, without introducing additional quantities andwithout the complications that other formulations have. Thisrepresentation is simple enough as to allow the analysis ofthe dynamics of quantum systems completely in phase space,formally as well as numerically, without recurring to coordi-nate representation. Now, a phase-space representation ofquantum mechanics leads to the question of the existence ofphase-space versions of the special functions normally usedin the coordinate space. In a previous paper, we have foundthe phase-space eigenfunctions for the case of the harmonicoscillator @1# and in this paper we look for the phase-spaceeigenfunctions for the quantum particle in a linear potential.
In Sec. II we introduce a set of two-variable special func-tions with properties similar, and that can be reduced, to theusual Airy functions. These functions are the solution to atwo-variable partial differential equation. In Sec. III a briefreview of the recently introduced phase-space representationof nonrelativistic quantum mechanics is made. In Sec. IV theeigenvalue problem for the phase-space quantum particle ina linear potential is solved. We find that some solutions re-semble the coordinate wave function and others resemble theclassical limit. Finally, in Sec. V some concluding remarksare made.
II. TWO-VARIABLE AIRY FUNCTION
The two-variable Airy function is defined by
Ai ~p,q;a!51
pe2 ip~q2 iap!/2E
2`
`
dteit3/32at2/21 i t ~q2 iap!
52
pe2 ip~q2 iap!/2E
0
`
dte2at2/2
3cosF t33 1t~q2 iap!G , ~1!
where 0<a<`. Note that whena→0, the above function,aside from a phase factor, becomes the usual one-variableAiry function. This function satisfies a second-order partialdifferential equation of particularly simple type. By differen-tiating under the sign of integration, we find that
S p22 i]
]qD2
Ai ~p,q;a!1S q21 i]
]pDAi ~p,q;a!
52 ie2 ip~q2 iap!/2@eit3/32at2/21 i t ~q2 iap!# t52`
t5` .
At the extremitiest56` the quantity in square bracketsvanishes. Therefore the equation
S p22 i]
]qD2
w1S q21 i]
]pDw50 ~2!
is satisfied byw5Ai( p,q;a). As we can see, the solution tothis equation is not unique, but it is a family of solutions thatdepend on the parametera. In Figs. 1 and 2 we show densityplots of the square magnitude and phase of the two-variableAiry function Eq. ~1!, for a51 anda50.05, respectively.For the square magnitude, darker regions indicate that thefunction has a large value and in the white region the func-tion is very small. For the phase plot, large negative valuesare indicated by white regions and the darkest regions indi-cate where the phase has the largest positive value.
Other equations for this function are
F S q21 i]
]pD2 iaS p22 i]
]qD GAi ~p,q;a!
5~q2 iap!Ai ~p,q;a!,
PHYSICAL REVIEW A JUNE 1996VOLUME 53, NUMBER 6
531050-2947/96/53~6!/3792~6!/$10.00 3792 © 1996 The American Physical Society
which looks like an ‘‘eigenvalue’’ equation with the eigen-valueq2 iap, and
]
]qRe Ai* ~p,q;a!S p22 i
]
]qDAi ~p,q;a!
2]
]puAi ~p,q;a!u250 , ~3!
which is a conservation equation for the ‘‘density’’uAi( p,q;a)u2.
Some of the properties of these functions can be easilyderived from the equalities obtained by integratingexp@iv3/32av2/21 iv(q2 iap)# around the contoursCishown in Fig. 3. The needed results are
E0
`
dt eit3/32at2/21 i t ~q2 iap!
52e22ip/3E2`
0
dt expF i3 t32 a
2e2ip/3t21 i t ~qe22ip/3
2 iae2ip/3pe2ip/3!G , ~4!
E2`
0
dt eit3/32at2/21 i t ~q2 iap!
52e2ip/3E0
`
dt expF i3 t32 a
2e22ip/3t21 i t ~qe2ip/3
2 iae22ip/3pe22ip/3!G , ~5!
FIG. 1. Density plots of the square magnitude and phase of thetwo-variable Airy function fora51.
FIG. 2. Density plots of the square magnitude and phase of thetwo-variable Airy function fora50.05.
FIG. 3. Contours used in the text.
53 3793SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE . . .
E0
`
dt e2t3/31at2/21t~q2 iap!
5e2 ip/6E0
`
dt expF i3 t32 a
2e2ip/3t21 i t ~qe22ip/3
2 iae2ip/3pe2ip/3!G , ~6!
E0
`
dte2t3/31at2/21t~q2 iap!
5eip/6E0
`
dt expF2i
3t32
a
2e22ip/3t22 i t ~qe2ip/3
2 iae22ip/3pe22ip/3!G . ~7!
A power-series expansion for Airy’s function is given by
Ai ~p,q;a!52
pe2 ip~q2 iap!/2(
n50
`1
n!
3~q2 iap!nE0
`
dte2t3/323at2/10tn
3cosS 25at22np2
32
p
6 D .Now, Eq. ~2! is unaffected when (p,q) is replaced by(pe62p i /3,qe72p i /3); hence other solutions are obtained bymaking these replacements. However, the functions so ob-tained are not independent, but are related as
Ai ~p,q;a!1e22p i /3Ai ~pe2p i /3,qe22p i /3;ae2p i /3!
1e2p i /3Ai ~pe22p i /3,qe2p i /3;ae22p i /3!50 ,
as can be easily verified by combining Eqs.~4!–~7!.Another solution, which we denote by Bi(p,q;a), to Eq.
~2! is defined by the combination
Bi~p,q;a![e2 ip/6Ai ~pe2p i /3,qe22p i /3;ae2p i /3!
1eip/6Ai ~pe22p i /3,qe2p i /3;ae22p i /3!
51
pe2 ip~q2 iap!/2E
0
`
dtH e2t3/31at2/21t~q2 iap!
1e2at2/2sinF t33 1t~q2 iap!G J . ~8!
In Figs. 4 and 5 we show density plots of the square magni-tude and phase of the second solution Eq.~8! for the casesa51 and a50.05, respectively. These functions increaserapidly in the regionq.0. As for the case of the functionsAi( p,q;a), by replacing (p,q) with (pe62p i /3,qe72p i /3) weobtain another two solutions. However, these solutions arenot independent; in fact,
Bi~p,q;a!1e22p i /3Bi~pe2p i /3,qe22p i /3;ae2p i /3!
1e2p i /3Bi~pe22p i /3,qe2p i /3;ae22p i /3!50 .
One last relationship between these functions is
Ai ~pe72p i /3,qe62p i /3;ae72p i /3!
51
2e6p i /3@Ai ~p,q;a!7Bi~p,q;a!#.
III. QUANTUM-MECHANICAL PHASE SPACE
In this paper, we make use of the recently introducedphase-space representation of non-relativistic quantum me-chanics@1,2# in which the operators associated with the mo-mentumP, coordinateQ, and inverse coordinateQ21 op-erators are given by
P↔p
22 i\
]
]q, Q↔
q
21 i\
]
]p,
Q21↔2i
\eipq/2\E
2`
p
dpe2 ipq/2\.
The operatorsP↔p/22 i\]/]q and Q↔q/21 i\]/]p donot commute with each other; in fact,@Q,P#5 i\ and so far
FIG. 4. Density plots of the square magnitude and phase of thesecond kind two-variable Airy function Bi(p,q;a), for a51.
3794 53GO. TORRES-VEGAet al.
this formulation has shown to comply with the requirementsfor a quantum representation. The basis vectors areuG&,where G5(p,q) denotes a point in phase space, and theprojectionc(G)5^Guc& is the phase-space wave function,with its complex conjugate given by the projectionc* (G)5^cuG&. The quantity uc(G)u2 is, by definition, anon-negative quantity that can be considered to be the quan-tum probability density in phase space. The inner productbetween two phase-space functionsc(G) andf(G) is de-fined in the usual waycuf&5*c* (G)f(G)dG.
Then, the phase-space Schro¨dinger equation is given by
i\]
]t^Guc&5
1
2m F S p22 i\]
]qD2
1VS q21 i\]
]pD G^Guc&,
whereV(q/21 i\]/]p) is the potential function evaluated atq/21 i\]/]p. Within this representation one can analyze,formally as well as numerically, quantum dynamics entirelyin phase space in the same way as it is done in coordinate orabstract spaces, without introducing additional quantities andcomplications into the theory.
For instance, finding eigenvalues and eigenfunctions ofthe Hamiltonian operator can be done in a way similar tohow it is done in coordinate representation: by analyticallysolving the eigenvalue problem
F 1
2m S p22 i\]
]qD2
1VS q21 i\]
]pD G^GucE&5E^GucE&
or numerically, by diagonalizing the matrix elements of theHamiltonian operator, in a given basis, or by propagating anonstationary initial phase-space wave function^Guc0& andutilizing the standard time-dependent formalism, which re-quires the evaluation of the Fourier transformslimT→`*2T
T dt exp(ivt)^c0uct& and
^GucE&} limT→`
1
2TE2T
T
dt eiEt/\^Guc t&.
One can recover the usual coordinate or momentumspaces from the phase-space representation. The wave func-tion in coordinate spacequc& can be recovered from thewave function in phase space^Guc& by means of the projec-tion ^quc&5*2`
` exp(ipq/2\)^Guc&dp and the momentumwave function is obtained from the wave function inphase space by means of the projectionpuc&5*2`
` exp(2ipq/2\)^Guc&dq. Similarly, the phase-spacematrix elementsp,quOup8,q8& of an operatorO can be pro-jected to coordinate or momentum spaces by using theequalities
^quOuq&51
2p\E2`
`
duE2`
`
dse2 iqs/\^u2s,quOuu1s,q&
and
^puOup&51
2p\E2`
`
duE2`
`
dse2 ips/\^p,u1suOup,u2s&.
The diagonal matrix element of the quantum probabilityconservation equation is
]
]t^GuruG&52
]
]q
1
2m@^GuPruG&1^Gur PuG&#
1]
]p F (n5M
1
V2n(l51
n
^GuQ2 l rQ2n1 l21uG&
1 (n51
`
Vn(l50
n21
^GuQl rQn2 l21uG&G , ~9!
wherer is the time-dependent density operator and we haveassumed that the potential function can be written asV(q)5(n52M
` Vnqn for some integerM . Note that in the
above equation, the termV0 is no longer there and it iscombining the corresponding equations in coordinate
]
]t^quruq&52
]
]q
1
2m@^quPruq&1^qur Puq&#
and momentum
FIG. 5. Density plots of the square magnitude and phase of thesecond kind two-variable Airy function Bi(p,q;a), for a50.05.
53 3795SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE . . .
]
]t^purup&5
]
]p F (n5M
1
V2n(l51
n
^puQ2 l rQ2n1 l21up&
1 (n51
`
Vn(l50
n21
^puQl rQn2 l21up&Gspaces, 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 alinear potential of strengthK is
F 1
2m S p22 i\]
]qD2
1KS q21 i\]
]pD GcE~G!5EcE~G!.
This equation can be put in the form of Eq.~2! by makingthe replacementsp→p(2mK\)1/3 and q→(\2/2mK)1/3q12E/K. Then the eigenfunction, with eigenvalueE, for theparticle in a linear potential is given by
cE~G;a!5NAi F p
~2mK\!1/3,S 2mK
\2 D 1/3S q22E
K D ;abG ,~10!
whereN is the normalization factor. We have chosen thefunction Ai(p,q;a), instead of Bi(p,q;a), because it is thenondivergent solution.
In Fig. 6, there are plots of the dimensionless probabilitydensityuAi( p,q;a)u2, Eq. ~1!, and of the corresponding flux@see Eqs.~3! and ~9!# for a50.01,1,30. As can be seen inthis figure , for small value ofa50.01, the square magnitudeuAi( p,q;a)u2 resembles the usual quantum coordinate prob-ability for this system in theq direction, whereas it is verybroad in p. The solid line is the classical trajectoryp21q50, and we can notice that, for this value ofa, thereis a lot of interference~or tunneling between the twobranches of the classical trajectory! where the probability islarge.
For a51 ~see Fig. 6!, the coordinateq and momentump have the same weight inuAi( p,q;a)u2 and the probabilitybegins to be centered around the classical trajectory
FIG. 6. Phase-space probability and flux densities for the quan-tum particle in a linear potential. Casesa50.01,1,30. The solid lineis the classical trajectoryp21q50.
FIG. 7. Classical stationary density exp@2(p21q)/30#.
FIG. 8. Wigner function equivalent to the probability densitiesin Fig. 6. The solid line is the classical trajectoryp21q.
3796 53GO. TORRES-VEGAet al.
p21q50. The interference~or tunneling! has decreasedsubstantially with respect to the casea50.01.
For the large valuea530 ~see Fig. 6!, the probability isdefinitely centered around the classical trajectory and the in-terference is gone, imitating classical behavior. In order tosee how far from a classical stationary density is the quantumdensity fora530, in Fig. 7 there is a density plot of theclassical stationary density exp@2(p21q)2/30#. These plotsare almost identical.
Thus the functions introduced in this paper can be used asa stationary solution in a variety of situations ranging fromthe quantum coordinatelike~small value ofa) to the quan-tum phase-space-like~equal weight for p and q, i.e.,a51) to a classical-like~large value ofa) density. In thiscase, the classical limit corresponds to a large value of thefree parametera.
V. CONCLUDING REMARKS
We can make a comparison between the above results andother phase-space functions. For instance, in Fig. 8 we showa density plot of the Wigner function
W~p,q!522/3
p2 Ai @22/3~p21q!#, ~11!
where
Ai ~x!51
pE2`
`
eit3/31 ixtdt
is the usual Airy function, which is the solution to the ordi-nary differential equation
d2
dx2w~x!1xw~x!50 .
This is the equivalent to the calculation shown in Fig. 6. Asis usual for Wigner functions, this density is not always posi-tive and has peaks at many places aside from near the clas-sical trajectory, with ‘‘wild’’ oscillatory behavior, hinderingthe identification of correlation between coordinate and mo-menta.
There is a big difference between the functions introducedin Sec. II and Eq.~11!. The Wigner function Eq.~11! is justthe usual Airy function, which is the solution to an ordinarydifferential equation—the Schro¨dinger equation in coordi-nate space—with argument 22/3(p21q). In contrast, thefunctions of Sec. II are solutions to a two-variable partialdifferential equation: a Schro¨dinger equation in phase space.The functions used in this paper seem to be appropriate for afull quantum phase-space analysis, which includes a coordi-natelike wave function as well as a classical-like wavepacket, depending on the value of a free parameter.
The introduction of orthogonal polynomials and specialfunctions in phase space shows that it is possible to deal withquantum systems completely in phase space, in a fashionsimilar to how it is done in abstract, coordinate, or momen-tum spaces. This means that one can have quantum wavefunctions that depend on both coordinate and momentumvariables without violating any of the quantum principlesand allows us to use the methods of standard nonrelativisticquantum mechanics. All of these things can be done in avery simple manner without the complications found in otherformulations.
ACKNOWLEDGMENTS
We would like to acknowledge financial support fromCONACyT and SNI, Mexico.
@1# Go. Torres-Vega and John H. Frederick, J. Chem. Phys.93,8862 ~1990!; 98, 3103 ~1993!; Phys. Rev. Lett.67, 2601~1991!; Go. Torres-Vega, J. Chem. Phys.98, 7040~1993!; 99,1824~1993!; Go. Torres-Vega and J.D. Morales-Guzma´n, ibid.101, 5847~1994!.
@2# Qian-Shu Li and Xu-Guang Hu, Phys. Scr.51, 417 ~1995!.
@3# E. Wigner, Phys. Rev.40, 749 ~1932!; M. Hillery, R.F.O’Connell, M.O. Scully, and E.P. Wigner, Phys. Rep.106, 121~1984!, and references therein.
@4# K. Husimi, Proc. Phys. Math. Soc. Jpn.22, 264 ~1940!; Co-herent States, edited by J.R. Klauder and B.S. Skagerstum~World Scientific, Singapore, 1985!, and references therein.
53 3797SPECIAL FUNCTIONS AND QUANTUM MECHANICS IN PHASE . . .