Chattaraj P.K. (Ed.) Quantum Trajectories

Подождите немного. Документ загружается.

64 Quantum Trajectories

100 200 300 400 500 600

CG step

–0.04

–0.02

0.00

0.02

0.04

Node position

FIGURE 4.4 Location of excited state node for the last 600 CG steps. (Reproduced with

permission from Bittner, E. R., Maddox, J. B., and Kouri, D. J., J. Phys. Chem. A, 2009, http://

dx.doi.org/10.1021/jp9058017.)

gradient of V

. We have tested this approach on a number of other one-dimensional

bound-state problems with similar success.

4.5 EXTENSION OF SUSY TO MULTIPLE DIMENSIONS

While SUSY-QM has also been explored for one-dimensional, non-relativistic quan-

tum mechanical problems [3,27–31], thus far these studies have focused on the formal

aspects and on obtaining exact, analytical solutions for the ground state for speciﬁc

classes of problems. In several recent papers [15, 25, 32, 33], we have begun explor-

ing the SUSY-QM approach as the basis of a general computational scheme for

bound-state problems. Our initial studies have been restricted to one-dimensional

systems (for which there are, obviously, many powerful computational methods). In

our ﬁrst paper, we found that SUSY-QM (combined with a new periodic version of the

Heisenberg–Weyl algebra) yields a robust, natural way to treat an inﬁnite family of

hindered rotors [33]. Next we showed that the SUSY-QM leads to a general treatment

of an inﬁnite family of an harmonic oscillators (HO), such that highly accurate excited

state energies and wave functions could be obtained variationally using signiﬁcantly

smaller basis sets than a traditional variational approach requires [15]. Most recently,

we have considered a one-dimensional double well potential in which we solved for

the ground-state energy and wave function using a VQMC approach. Then using

SUSY-QM, we (numerically) generated an auxiliary Hamiltonian whose nodeless

ground state is isospectral (degenerate) with the ﬁrst excited state of the original sys-

tem Hamiltonian. This ground state was also easily determined by VQMC, yielding

excellent accuracy for the ﬁrst excited state energy [25]. Even more signiﬁcant, by

using the charge operators naturally generated in the SUSY-QM approach, we also

Super-Symmetric Quantum Mechanics 65

obtained excellent accuracy for the ﬁrst excited state wave function. Furthermore, at

no point did we impose a ﬁxed node or symmetry on the excited state wave function

and our calculation only involved working with a nodeless ground state.

Of course, all this begs the question: Can this approach be generalized to higher

numbers of dimensions and to more than a single particle? There has been substantial

effort in the past to do just this [3,28–30,34–45]. However, to date, no such general-

ization has been found that is able to generate all the excited states and energies even

for so simple a system as a pair of separable, one-dimensional HO or equivalently, for

a separable two-dimensional single HO. In our most recent, unpublished work [32],

we have succeeded in obtaining such a generalization and showed that it does, in fact,

yield the correct analytical results for separable and non-separable problems. In the

next section, we present a succinct summary of our approach. The major question

now is whether this formalism provides a basis for a robust, computational method

for determining excited state energies and wave functions for large, strongly corre-

lated systems using either QMC or variational algorithms applied solely to nodeless

ground-state problems.

4.5.1 D

IFFICULTIES IN EXTENDING BEYOND ONE DIMENSION

To move beyond one-dimensional SUSY, Ioffe and coworkers have explored the use

of higher-order charge operators [42,43,46,47], and Kravchenko has explored the use

of Clifford algebras [48]. Unfortunately, this is difﬁcult to do in general, the reason

being that the Riccati factorization of the one-dimensional Schrödinger equation does

not extend easily to higher dimensions. One remedy is to write the charge operators

as vectors

A = (+

∂ +

W ) and with

= (−

∂ +

W )

†

as the adjoint charge operator.

The original Schrödinger operator is then constructed as an inner product

A. (4.39)

Working through the vector product produces the Schrödinger equation

φ = (−∇

+ W

− (

∇·

W ))φ = 0 (4.40)

and a Riccati equation of the form

U(x) = W

−

∇·

W . (4.41)

For a two-dimensional HO, we would obtain a vector superpotential of the form

W =−

(1)

∇ψ

(1)

(

x, y

)

= (W

, W

). (4.42)

Let us look more closely at the

∇·

W part. If we use the form that

W =−

∇ ln ψ,

then −

∇·

∇ ln ψ =−∇

ln ψ which for the two-dimensional oscillator results in

∇·

W = 2. Thus,

−

∇·

W = (x

+ y

) − 2 (4.43)

66 Quantum Trajectories

which agrees with the original symmetric harmonic potential. Now, we write the

scaled partner potential as

= W

∇·

W = (x

+ y

) + 2. (4.44)

This is equivalent to the original potential shifted by a constant amount

= U

+ 4. (4.45)

The ground state in this potential would have the same energy as the states of the

original potential with quantum numbers n + m = 2. Consequently, even with this

naïve factorization, one can in principle obtain excitation energies for higher dimen-

sional systems, but there is no assurance that one can reproduce the entire spectrum

of states.

The problem lies in the fact that neither Hamiltonian H

nor its associated potential

is given correctly by the form implied by Equations 4.40 and 4.44. Rather, the

correct approach is to write the H

Hamiltonian as a tensor by taking the outer product

of the charges

rather than as a scalar

A ·

. At ﬁrst this seems unwieldy

and unlikely to lead anywhere since the wave function solutions of

ψ = E

ψ (4.46)

are now vectors rather than scalars. However, rather than adding undue complexity to

the problem, it actually simpliﬁes matters considerably. As we shall demonstrate in a

forthcoming paper, this tensor factorization preserves the SUSY algebraic structure

and produces excitation energies for any n-dimensional SUSY system. Moreover,

this produces a scalar → tensor → scalar hierarchy as one moves to higher excita-

tions [32].

4.5.2 V

ECTOR SUSY

We now give a brief summary of our new generalization of SUSY-QM to treat higher

dimensionality and more than one particle. Previous attempts generally involved

introducing additional, “spin-like” degrees of freedom [29,30, 37–40, 42, 43, 46, 47,

49]. In our approach, we make use of a vectorial technique that can deal simultaneously

with either higher dimensions or more than one particle. In fact, the two problems

are dealt with in exactly the same manner. Therefore, for simplicity, we consider a

general n-dimensional distinguishable particle system with orthogonal coordinates

}. The Hamiltonian is given by

H =−∇

+ V

, ...,x

) (4.47)

and the nodeless ground state satisﬁes the Schrödinger equation,

H ψ

(1)

= E

(1)

. (4.48)

Our units are such that 

/2m = 1.

Super-Symmetric Quantum Mechanics 67

We now deﬁne a “vector superpotential,”

, with components

1µ

=−

∂

∂x

ln ψ

(1)

. (4.49)

Then it is easily seen that the original Hamiltonian can be recast as

= (−∇ +

) · (∇+

) =

(4.50)

where the

and

are multi-dimensional generalizations of the SUSY charge

operators from Equation 4.39. This deﬁnes our “sector-1” (or “boson”) Hamiltonian

and Equation 4.48 can be written as

(1)

= E

(1)

. (4.51)

One can show that the vector superpotential is related to the original (scalar) poten-

tial via:

−∇·

. (4.52)

The various components of the charge operators,

and

, are deﬁned by

1µ

∂

∂x

+ W

1µ

and A

1µ

=−

∂

∂x

+ W

1µ

. (4.53)

Note that since these are associated with orthogonal degrees of freedom, the charge

operators can be applied either by individual components or in vector form.

Next, consider the Schrödinger equation for the ﬁrst excited state of H . We can

write this using the charge operators as

(1)

= E

(1)

= (

+ E

(1)

)ψ

(1)

. (4.54)

We apply

to Equation 4.9:

(

) ·

(1)

= (E

(1)

− E

(0)

)

(1)

. (4.55)

Here we identify (

†

) as a new, auxiliary Hamiltonian. It is important to note that

this is constructed from the outer or tensor product of the charge operators rather

than from inner or dot product as used in constructing H

. Its eigenvector,

(1)

is isospectral with the excited state, ψ

(1)

of H

(since E

(1)

is known, determining

(1)

−E

(0)

) yields E

(1)

). We therefore deﬁne the tensor Hamiltonian for the second

sector as

←→

†

(4.56)

In analogy with the original descriptions of SUSY, we refer to the partner pairs as “boson” and “fermion”

sectors or less poetically as “sector-1,” “sector-2,” and so forth.

68 Quantum Trajectories

and vector state function as

(2)

(1)

− E

(0)

)

(1)

. (4.57)

It is easy to show that the ground-state energy of

←→

is related to the ﬁrst excitation

energy of the original Hamiltonianm

(2)

= E

(1)

− E

(0)

. (4.58)

Furthermore, the ground state of

←→

is also nodeless. This has been explicitly shown

to be true for the separable two-particle HOs considered earlier [32]. Therefore, we

propose to apply both the VQMC and the standard variational methods to determine

(2)

and

(2)

. Of course, knowing the second sector ground-state energy also gives us

the ﬁrst excited state energy of the original Hamiltonian (Equation 4.58). Furthermore,

we form the scalar product of

←→

(2)

= E

(2)

(4.59)

with

obtaining

(

)

(2)

= E

(2)

. (4.60)

Clearly, this is exactly

(

(2)

) = E

(1)

(

(2)

) (4.61)

so we can conclude that

(1)



(2)

(

(2)

). (4.62)

Thus we also obtain the excited state wave function without any signiﬁcant addi-

tional computational effort. This is because applying the charge operator is much

simpler than solving an eigenvalue problem (it is a strictly linear operation). Evi-

dence from our one-dimensional studies indicates that the accuracy of the excited

states obtained using the SUSY-QM charge operator is signiﬁcantly higher, for a

given basis set, than what is obtained variationally (or with QMC) from the origi-

nal Hamiltonian [15, 32]. This procedure can be continued as follows. We deﬁne a

sector-2 vector superpotential with components

2µ

∂

∂x

ln ψ

(2)

0µ

. (4.63)

Then it follows that

(2)

= (∇+

) · ψ

(2)

= 0 (4.64)

Super-Symmetric Quantum Mechanics 69

so we can write

←→

+ E

(2)

I (4.65)

and Equation 4.59 is still satisﬁed. We form the scalar product of

with the ﬁrst

excited state Schrödinger equation to obtain

(

)

(2)

= E

(2)

. (4.66)

Then we deﬁne the sector-3 scalar Hamiltonian by

+ E

(2)

(4.67)

with the ground-state wave equation

(3)

= E

(3)

. (4.68)

It is easily seen that E

(3)

= E

(2)

−E

. This procedure continues until all bound states

of the original Hamiltonian are exhausted. It should also be clear that the sector-2

excited state wave function is obtained from the nodeless sector-3 ground state by

applying

to it. Then the second excited state for sector-1 results from taking the

scalar product of

with

(2)

. The approach thus leads to an alternating sequence

of scalar and tensor Hamiltonians, but in all cases we need only determine nodeless

ground states.

There are two additional aspects of the tensor sector problem that require dis-

cussion. First we consider the validity of the Rayleigh–Ritz variational principle. It

is easily seen from Equation 4.56 that

←→

is a Hermitian operator. Therefore, its

eigenspectrum is real and its eigenvectors are complete. With these facts in hand, the

proof of the variational principle follows the standard one in every detail. This is also

true for the Hylleraas–Undheim theorem.

Second, the QMC method is also directly applicable to the tensor sector problem.

For the example discussed above, we note that the energy is given by

trial



dτ

trial

←→

trial



dτ

trial

. (4.69)

We next note that the integral can be expanded in terms of its components as

trial



µν



dτ(ψ

µ,trial

2,µν

ν,trial

)





dτ(ψ

µ,trial

)

. (4.70)

It is then clear that each separate integral can be evaluated by QMC. For example,

the µ = ν cross-term is divided and multiplied by

µ,trial



dτψ

ν,trial

µ,trial

. (4.71)

70 Quantum Trajectories

Then the sampling is done relative to the mixed probability distribution,

µν

µ,trial

ν,trial



dτψ

µ,trial

ν,trial

. (4.72)

A similar expression applies to each term in the energy expression and the evaluation

would need to be performed self-consistently.

Thus far, we have developed a formalism that appears to be suitable for extending

the SUSY-QM technique to higher-dimensional systems. We believe the approach we

have outlined above will provide the mathematical basis for a number of potentially

interesting theoretical results. Moreover, we anticipate that when combined with either

variational or Monte Carlo methods, our multi-dimensional extension of SUSY-QM

will facilitate the calculation of accurate excitation energies and excited state wave

functions.

4.6 OUTLOOK

I presented a number of avenues we are actively pursuing with the goal of using

SUSY-QM or SUSY-inspired-QM to solve problems that are difﬁcult to solve using

more conventional approaches. In addition to what I have discussed here we exploring

the use of the Riccati equation to solve quantum scattering problems. It is as if one

of the coauthors of this paper (DJK) has come full-circle since one of his ﬁrst papers

concerned solving the Hamilton–Jacobi equation for the action integral in quantum

scattering [50],

iS/ =−



W (r



)dr



The integrand in this last equation is the SUSY superpotential. Furthermore, there is

a connection between our work and the complex-valued quantum trajectories studied

by Wyatt and Tannor and their respective coworkers.

ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation (ERB: CHE-

0712981) and the Robert A. Welch foundation (ERB: E-1337, DJK: E-0608).

BIBLIOGRAPHY

1. E. Witten, Nucl. Phys. B (Proc. Supp.) 188, 513 (1981).

2. E. Witten, J. Different. Geom. 17, 661 (1982).

3. F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).

Coincidentally, Ref. [50] appeared in the J. Chem. Phys. issue immediately before the birthday of the

other author of this paper. There appears to be some interesting Karma at work here.

Super-Symmetric Quantum Mechanics 71

4. E. A. Hylleraas and B. Undheim, Z. Phys. 65, 759 (1930).

5. B. L. Hammond, W. A. Lester, and P. J. Reynolds, Monte Carlo Methods in Ab Initio

Quantum Chemistry,vol.1ofWorld Scientiﬁc Lecture and Cousre Notes in Chemistry

(World Scientiﬁc Publishing, River Edge, NJ, 1994).

6. A. R. Porter, O. K. Al-Mushadani, M. D. Towler, and R. J. Needs, J. Chem. Phys. 114,

7795 (2001), http://link.aip.org/link/?JCP/114/7795/1.

7. J. D. Doll, R. D. Coalson, and D. L. Freeman, J. Chem. Phys. 87, 1641 (1987).

8. R. J. Needs, P. R. C. Kent, A. R. Porter, M. D. Trowler, and G. Rajagopal, Int. J. Quantum

Chem. 86, 218 (2001).

9. S. Sorella, Phys. Rev. B 71, 241103 (2005).

10. D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, Phys. Rev. E 55, 3664 (1997).

11. T. Bouabça, N. B. Amor, D. Maynau, and M. C. ffarel, J. Chem. Phys. 130, 114107 (2009),

http://link.aip.org/link/?JCP/130/114107/1.

12. X. Oriols, J. J. García, F. Martín, J. Suñé, T. Gonzàlez, J. Mateos, and D. Pardo, App.

Phys. Lett. 72, 806 (1998).

13. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985), http://

link.aip.org/link/?JCP/82/1400/1.

14. J. C. Light, in Time-Dependent Quantum Molecular Dynamics, edited by J. Broeckhove

and L. Lathouwers (Plenum, New York, 1992), pp. 185–199.

15. D. J. Kouri, T. Markovich, N. Maxwell, and E. R. Bittner, J. Phys. Chem. A (2009), http://

dx.doi.org/10.1021/jp905798m.

16. J. B. Maddox and E. R. Bittner, J. Chem. Phys. 119, 6465 (2003), http://link.aip.org/

link/?JCP/119/6465/1.

17. D. Bohm, Phys. Rev. 85, 180 (1952).

18. P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge,

1993).

19. C. L. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999).

20. E. R. Bittner and R. E. Wyatt, J. Chem. Phys. 113, 8888 (2000).

21. R. E. Wyatt, c. L. Lopreore, and G. Parlant, J. Phys. Chem. 114, 5113 (2001).

22. S. W. Derrickson and E. R. Bittner, J. Phys. Chem. A 110, 5333 (2006), http://pubs.

acs.org/cgi-bin/article.cgi/jpcafh/2006/110/i16/pdf/jp055889q.pdf.

23. S. W. Derrickson and E. R. Bittner, J. Phys. Chem. A 111, 10345 (2007), http://

pubs.acs.org/doi/pdf/10.1021/jp0722657, http://pubs.acs.org/doi/abs/10.1021/jp0722657.

24. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), vol. 3

of Course of Theoretical Physics (Pergammon, Oxford, 1974), 3rd edn.

25. E. R. Bittner, J. B. Maddox, and D. J. Kouri, J. Phys. Chem. A (2009), http://

dx.doi.org/10.1021/jp9058017.

26. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985).

27. H. Baer, A. Belyaev, T. Krupovnickas, and X. Tata, Phys. Rev. D 65, 075024 (pages 8)

(2002), http://link.aps.org/abstract/PRD/v65/e075024.

28. A. A. Andrianov, N. V. Borisov, M. V. Ioffe, and M. I. Eides, Theoret. Math. Phys. 61, 965

(1984).

29. A. A. Andrianov, N. V. Borisov, M. I. Eides, and M. V. Ioffe, Phys. Lett. A 109, 143 (1985).

30. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. A 105, 19 (1984).

31. A. Gangopadhyaya, P. K. Panigrahi, and U. P. Sukhatme, Phys. Rev. A 47, 2720 (1993).

32. E. R. Bittner, D. J. Kouri, K. Maji, and T. J. Markovich, J. Phys. Chem., submitted (2010).

33. D. J. Kouri, T. Markovich, N. Maxwell, and B. G. Bodman, J. Phys. Chem. A 113, 7698

(2009).

72 Quantum Trajectories

34. P. T. Leung, A. M. van den Brink, W. M. Suen, C. W. Wong, and K. Young, J. Math. Phys.

42, 4802 (2001), http://link.aip.org/link/?JMP/42/4802/1.

35. R. de Lima Rodrigues, P. B. da Silva Filho, and A. N. Vaidya, Phys. Rev. D 58, 125023

(pages 6) (1998), http://link.aps.org/abstract/PRD/v58/e125023.

36. A. Contreras-Astorga and D. J. F. C., AIP Conference Proceedings 960, 55 (2007)

37. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theor

et.

Math. Phys. 72, 748 (1987).

38. A. A. Andrianov and M. V. Ioffe, Phys. Lett. B 205, 507 (1988).

39. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theoret. Math. Phys. 61, 1078 (1984).

40. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, JETP Lett. 39, 93 (1984).

41. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. B 181, 141 (1986).

42. F. Cannata, M. V. Ioffe, and D. N. Nishnianidze, J. Phys. A 35, 1389 (2002),

http://stacks.iop.org/0305-4470/35/1389.

43. A. Andrianov, M. Ioffe, and D. Nishnianidze, Phys. Lett. A 201, 103 (2002).

44. A. Das and S. A. Pernice, arXiv:hep-th/9612125v1 (1996).

45. M. A. Gonzalez-Leon, J. M. Gullarte, and M. de la Torre Mayado, SIGMA 3, 124 (2007).

46. A. A. Andrianov, M. V. Ioffe, and V. P. Spiridonov, Phys. Lett. A 174, 273 (1993).

47. A. A. Andrianov, M. V. Ioffe, and D. N. Nishnianidze, Theoret. Math. Phys. 104, 1129

(1995).

48. V. V. Kravchenko, J. Phys. A 38, 851 (2005).

49. R. I. Dzhioev and V. L. Korenev, Phys. Rev. Lett. 99, 037401 (2007), ISSN 0031-9007

(Print).

50. D. J. Kouri and C. F. Curtiss, J. Chem. Phys. 43, 1919 (1965), http://link.aip.org/

link/?JCP/43/1919/1.

Quantum Field Dynamics

from Trajectories

Peter Holland

CONTENTS

5.1 Introduction.....................................................................................................73

5.2 Spin and Trajectories.......................................................................................74

5.3 Trajectory Construction of the Wave Equation in a Riemannian Manifold....76

5.3.1 Lagrangian Picture..............................................................................76

5.3.2 Initial Conditions ................................................................................78

5.3.3 Eulerian Picture ..................................................................................79

5.4 Bosons.............................................................................................................81

5.5 Fermions..........................................................................................................82

5.6 Conclusion ......................................................................................................85

Bibliography ............................................................................................................85

5.1 INTRODUCTION

Transcending its origin in the debate on the interpretation of quantum theory, the con-

tinuous deterministic trajectory has proved a potent tool in computational quantum

mechanics (an excellent introduction to numerical trajectory techniques is provided

by Wyatt’s text [1]). It is straightforward to develop a method to derive the time-

dependent Schrödinger equation from a single-valued continuum of spacetime tra-

jectories, and supply a corresponding exact formula for the wavefunction [2].Anatural

language for this theory of evolution is offered by the hydrodynamic analogy, in which

wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and

the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum

ﬂuid may be developed from a variational principle involving a speciﬁc interaction

potential (the quantum internal potential energy), and the Euler–Lagrange equations

imply a nonlinear partial differential equation to calculate the trajectories of the ﬂuid

particles as functions of their initial coordinates and the initial wavefunction. The

latter supplies two sets of data to the particle model: the initial density (via the ampli-

tude) and the initial velocity (via the phase). The time-dependent wavefunction is then

computed via the standard map between the Lagrangian coordinates and the Eulerian

ﬁelds.

Much of the work in this area has concentrated on developing techniques in

the arena of non-relativistic spin 0 quantum mechanics, using trajectory ﬂows in