A simple approach to phase-space representation of quantum state vectors using the displacement-operator formalism is presented. Although the resulting expressions for the fundamental operators (position and momentum) are equivalent to those obtained by other methods, this approach provides both alternative mathematical foundation as well as physical interpretation of phase-space representation of quantum state vectors

Møller, Klaus B.

Caltech Authors - Main

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 5 MAY 1999
DownloadedComment on phase-space representation of quantum
state vectors
Klaus B. Mo llera)
Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology,
Pasadena, California 91125
~Received 13 July 1998; accepted for publication 1 February 1999!
A simple approach to phase-space representation of quantum state vectors using the
displacement-operator formalism is presented. Although the resulting expressions
for the fundamental operators ~position and momentum! are equivalent to those
obtained by other methods, this approach provides both alternative mathematical
foundation as well as physical interpretation of phase-space representation of quan-
tum state vectors. © 1999 American Institute of Physics.
@S0022-2488~99!01605-9#
I. INTRODUCTION
Over the past few years there has been a renewed interest in phase-space descriptions of
quantum systems. In a recent paper by Ban1 a novel approach to phase-space representation of
quantum state vectors is obtained within the relative-state formulation, and in this Comment we
make a few remarks on the physical contents of this construction. Also, we relate it to another
approach to phase-space representation of quantum state vectors, the so-called displacement-
operator approach.
The idea of phase-space representation of quantum state vectors, i.e., representation of a
quantum state as a probability amplitude depending on two real variables related to the position
and momentum coordinates goes back to the works of Fock2 and Bargmann.3 In their formulation,
a quantum state is represented as a complex function depending on one complex coordinate whose
real and imaginary part is proportional to the position and momentum coordinate, respectively.
This is a result of regarding the bosonic creation and annihilation operators as the fundamental
operators.
The relative-state formulation, on the other hand, treats the position and momentum operators
themselves as the fundamental operators and is therefore more closely related to the works of
Torres-Vega and Frederick4 and Harriman.5 Both of these works rely to a certain extent on Dirac’s
representation theory of quantum mechanics,6 either as a Hilbert-space-vector approach postulat-
ing the existence of a complete set of states depending on two real parameters that can be used as
a basis in phase space or a linear transformation onto phase space from position or momentum
space.
In fact, the relative-state representation of Ban1 becomes, under certain conditions, equivalent
to those of Torres-Vega and Frederick4 and Harriman.5 The relative-state formulation may there-
fore serve as a mathematical and physical foundation for the representations presented by these
authors since it is derived from first principles without assumptions or transformations from other
representations.
However, the relative-state formulation is not the only way to construct a phase-space repre-
sentation of quantum state vectors from first principles that becomes equivalent to those of Torres-
Vega and Frederick4 and Harriman.5 Below, we present an alternative construction, using the
displacement operators, and discuss the mathematical and physical differences between this
method the relative-state approach.
a!Electronic mail: klaus@cco.caltech.edu25310022-2488/99/40(5)/2531/5/$15.00 © 1999 American Institute of Physics
 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
2532 J. Math. Phys., Vol. 40, No. 5, May 1999 Klaus B. Mo ller
DownloadedThe displacement-operator approach is essentially equivalent to the coherent-state formalism
as put forward by, for instance, Klauder and Skagerstam7 and studied in some detail by the present
author.8 Hence, the presentation given here is extracted from these earlier works and put in a form
relevant for the present discussion. For a thorough review and analysis of the use of displacement
operators, the reader is referred to Refs. 7 and 8 and the references therein.
II. RELATIVE-STATE FORMULATION
The relative-state formulation is presented in great detail by Ban1 and here we only include a
few results relevant for the further discussion. The key of this approach is to enlarge the Hilbert
space H of a quantum system by introducing an auxiliary ~reference! quantum system and treat
quantum state vectors in the extended Hilbert space H˜ 5H^Hr , where Hr is the Hilbert space of
the reference system. A state vector in the extended Hilbert space H˜ then becomes uC&&[uc&
^ uf&r where uf&r is the reference state.
A set of state vectors $uv(r ,k;s)&&ur ,kPR% may be introduced1 that becomes a complete
orthonormal system in H˜ . These state vectors, which can be written on the following form:
uv~r ,k;s !&&[
1
A2p
e2i~11s !kr/2E
2`
`
dx eikxux& ^ ux2r&r ~1!
~as in Ref. 1, we set \51 throughout this Comment! are simultaneous eigenstates of the operators
xˆ2 xˆ r , and pˆ1 pˆ r ,
~ xˆ2 xˆ r!uv~r ,k;s !&&5ruv~r ,k;s !&&, ~2!
~ pˆ1 pˆ r!uv~r ,k;s !&&5kuv~r ,k;s !&&. ~3!
However, when we investigate the properties of the relevant quantum system, we only need a
description of this system in the Hilbert space H. Thus, the extended Hilbert space is reduced
again by fixing the state vector of the reference system. For any fixed state vector uf&r of the
reference system, the set $uv(r ,k;s)&ur ,kPR%, where
uv~r ,k;s !&[r^fuv~r ,k;s !&&5
1
A2p
e2i~11s !kr/2E
2`
`
dx eikxf*~x2r !ux&, ~4!
becomes an overcomplete system in the Hilbert space H.1 Therefore, the relevant quantum system
can be represented by an L2(2) normalized wave function cv(r ,k;s)[^v(r ,k;s)uc& depending
of the two real parameters k and r. In this representation, the fundamental operators xˆ and pˆ take
the form
^v~r ,k;s !u xˆuc&5F12 ~11s !r1i ]]kGcv~r ,k;s !, ~5!
^v~r ,k;s !u pˆuc&5F12 ~12s !k2i ]]r Gcv~r ,k;s !. ~6!
Apart from some notational differences these are essentially the expressions given by Torres-Vega
and Frederick4 and Harriman5 in their representations. Thus, the construction by Ban1 may serve
as a mathematical foundation for the work of Torres-Vega and Frederick4 and Harriman.5 Fur-
thermore, the relative-state formulation provides a physical interpretation of the wave function
cv(r ,k;s) and the parameters k and r as phase-space coordinates. In light of Eqs. ~2! and ~3!, the
function ucv(r ,k;s)u2 represents the probability distributions of the eigenvalues of the operators
xˆ2 xˆ r and pˆ1 pˆ r in the extended Hilbert space H˜ . 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
2533J. Math. Phys., Vol. 40, No. 5, May 1999 Klaus B. Mo ller
DownloadedAlternatively, one may utilize ucv(r ,k;s)u2 as a combined probability distribution directly in
the r ,k-parametrized space as follows:
r¯[E
2`
`
drE
2`
`
dk rucv~r ,k;s !u25xc2xf , ~7!
k¯[E
2`
`
drE
2`
`
dk kucv~r ,k;s !u25pc1pf . ~8!
Here, xc5^cu xˆuc& and so on. Thus, r and k may be interpreted as phase-space coordinates in the
sense that the average value of r equals the relative position between the relevant and the reference
system, and the average value of k equals the sum of the momenta of the relevant and the
reference system. Hence, the physical interpretation of the wave function depends on the reference
state, although the operator expressions, Eqs. ~5! and ~6!, do not, and from this point of view the
most satisfactory representation is obtained using a reference state with ^fu xˆuf&50 and
^fu pˆuf&50. In general, also the physical interpretation of higher momenta of r and k depend on
the reference system.1
III. DISPLACEMENT-OPERATOR APPROACH
Here we present an alternative derivation from first principles of the phase-space representa-
tion of quantum state vectors that also becomes equivalent to the ones of Torres-Vega and
Frederick4 and Harriman5 and therefore to the result of Ban,1 as well. However, the derivation
presented here differs from the one obtained in the relative-state formulation in both the math-
ematical foundation and in the physical interpretation of the phase-space wave functions. In fact,
it resembles closely Dirac’s construction of the usual position and momentum representations.6
Two things are important for the definitions of these representations. First, the basis states,
denoted by ur&x and uk&p , are eigenstates of the position and momentum operator, respectively,
xˆur&x5rur&x and pˆuk&p5kuk&p . ~9!
Second, the position ~momentum! eigenstate with eigenvalue r(k) can be generated from the
eigenstate with eigenvalue r50 (k50) by a displacement operator,
ur&x5Dˆ x~r !u0&x and uk&p5Dˆ p~k !u0&p , ~10!
where the displacement operators are given as Dˆ x(r)5exp(2irpˆ) and Dˆ p(k)5exp(ikxˆ).6 The
wave function in position ~momentum! space is then obtained by projection onto a position
~momentum! eigenstate, c(r)[^ruc& (c(k)[^kuc&). This implies that the displacement opera-
tors when acting on a state displace the expectation value of the position or momentum by r and
k, respectively.
An identical approach to a phase-space representation of quantum state vectors would require
the existence of an Hermitian operator representing a point in phase space. Torres-Vega and
Frederick4 claim that such an operator exist but without proof and, in fact, the existence of such an
operator would violate the Heisenberg uncertainty relation. In the relative-state formulation, a
close resemblance is obtained for the basis states uv(r ,k;s)&& in the extended Hilbert space; cf.
Eqs. ~2! and ~3!.
Nevertheless, an r ,k-parametrized basis can be constructed utilizing displacement operators.
In general, a displacement operator that displaces the expectation values of the position and
momentum for any state by r and k simultaneously, can be defined as7,8
Dˆ s~r ,k !5exp@ i~kxˆ2r pˆ2skr/2!# , ~11! 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
2534 J. Math. Phys., Vol. 40, No. 5, May 1999 Klaus B. Mo ller
Downloadedwhere s is real number determining the phase such that Dˆ 1(r ,k)5Dˆ q(r)Dˆ p(k), Dˆ 21(r ,k)
5Dˆ p(k)Dˆ q(r), and Dˆ 0(r ,k) is a symmetric combination. An r ,k-parametrized state vector may
then be defined as uV(r ,k;s)&[(2p)21/2Dˆ s(r ,k)ux&, where ux& is an arbitrary normalized state,
and the set $uV(r ,k;s)&ur ,kPR% becomes an overcomplete set of normalized vectors.7 The set
$uV(r ,k;s)&ur ,kPR% can therefore be used as a basis and the relevant quantum system repre-
sented by the L2(2) normalized wave function cV(r ,k;s)[^V(r ,k;s)uc&, depending on the two
real parameters k and r. These basis vectors obviously satisfy the displacement relation
uV~r ,k;s !&5Dˆ s~r ,k !uV~0,0;s !&. ~12!
Using that
i
]
]k D
ˆ ~r ,k;s !5F12 ~11s !r2 xˆ GDˆ ~r ,k;s !, ~13!
i
]
]r
Dˆ ~r ,k;s !52F12 ~12s !k2 pˆ GDˆ ~r ,k;s !, ~14!
it is seen that in this representation, the fundamental operators xˆ and pˆ take the same form as in
the relative-state formulation, given by Eqs. ~5! and ~6!.
Therefore, the displacement-operator approach provides an alternative derivation from first
principles to the results obtained within the relative-state formalism. Here, the state of the relevant
system is projected onto an auxiliary ~reference! state ux&, displaced by r and k, whereas the
auxiliary state uf& in the relative-state formulation is utilized to project the orthonormal basis in
the extended Hilbert space onto a reduced Hilbert space. Thus, the auxiliary states play different
physical roles, as can also be seen from the relations
r¯[E
2`
`
drE
2`
`
dk rucV~r ,k;s !u25xc2xx , ~15!
k¯[E
2`
`
drE
2`
`
dk kucV~r ,k;s !u25pc2px . ~16!
Hence, r and k may here be interpreted as phase space coordinates, in the sense that the average
values obtained using ucV(r ,k;s)u2 as a combined probability distribution equal the relative po-
sition and momentum, respectively, between the relevant and the auxiliary system. Hence, the
displacement-operator approach provides a more symmetrical interpretation of the
r ,k-parametrized representation of the quantum state vector.
Since
Dˆ s~r ,k !ux&5e2i~11s !kr/2E
2`
`
dx eikxx~x2r !ux& , ~17!
we see that the displacement-operator approach and the phase-space representation obtained
within the relative-state formulation become formally identical if x(x)5f*(x); cf. Eq. ~4!, which
implies that pf52px , as expected @compare Eqs. ~8! and ~16!#.
In conclusion, we have shown that the two different mathematical approaches to a phase-
space representation of quantum state vectors lead to identical expressions for the fundamental
operators. However, usage of the well-known technique of displacement operators is in spirit
closer to the construction of the usual position and momentum representations and, also, it pro-
vides a more transparent physical interpretation of the auxiliary state as a ‘‘probe’’ state in phase 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
2535J. Math. Phys., Vol. 40, No. 5, May 1999 Klaus B. Mo ller
Downloadedspace.8 With this interpretation, phase-space representation of quantum state vectors becomes a
powerful tool and has been applied recently in the study of quantum dynamics directly in phase
space9 or as a route to semiclassical approximations.10
ACKNOWLEDGMENTS
The author thanks Professor Ahmed H. Zewail for his kind hospitality at Caltech and the
Carlsberg Foundation of Denmark for financial support.
1 M. Ban, J. Math. Phys. 39, 1744 ~1998!.
2 V. Fock, Z. Phys. 49, 339 ~1928!.
3 V. Bargmann, Commun. Pure Appl. Math. 14, 187 ~1961!.
4 Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 ~1993!.
5 J. E. Harriman, J. Chem. Phys. 100, 3651 ~1994!.
6 P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. ~Oxford University Press, Oxford, 1958!; W. H. Louisell,
Quantum Statistical Properties of Radiation ~Wiley, New York, 1973!, Chap. 1.
7 J. R. Klauder and B.-S. Skagerstam, Coherent States ~World Scientific, Singapore, 1985!.
8 K. B. Moller, T. G. Jorgensen, and Go. Torres-Vega, J. Chem. Phys. 106, 7228 ~1997!.
9 Go. Torres-Vega, K. B. Moller, and A. Zu´n˜iga-Segundo, Phys. Rev. A 57, 771 ~1998!.
10 S. Jang, M. Zhao, and S. A. Rice, Chem. Phys. 230, 237 ~1998!. 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp


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