PHASE SPACE APPROACH TO SOLVING THE SCHRÖDINGER EQUATION: THINKING INSIDE THE BOX

DAVID J. TANNOR, NORIO TAKEMOTO, and ASAF SHIMSHOVITZ

Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel

CONTENTS

1. I. Introduction
2. II. Theory
   1. A. von Neumann Basis on the Infinite Lattice
   2. B. Fourier Method
   3. C. The Periodic von Neumann Basis (pvN)
   4. D. Biorthogonal von Neumann Basis Set (bvN)
   5. E. Periodic von Neumann Basis with Biorthogonal Exchange (pvb)
3. III. Application to Ultrafast Pulses
4. IV. Applications to Quantum Mechanics
   1. A. Time-independent Schrödinger Equation (TISE)
      1. 1. Formalism
      2. 2. 1D Applications
      3. 3. Multidimensional Applications
      4. 4. Scaling of the Method with h and with Dimensionality
      5. 5. Wavelet Generalization
   2. B. Time-dependent Schrödinger Equation (TDSE)
5. V. Applications to Audio and Image Processing
6. VI. Conclusions and Future Prospects
7. Acknowledgments
8. References

I. INTRODUCTION

In 1946, Gabor proposed using a set of Gaussians located on a time-frequency lattice as a basis for representing arbitrary signals [1]. Gabor's motivation can be understood by considering Fig. 1a. If one considers an acoustical signal, generally there is some form of time-frequency correlation. This is made explicit in musical notation, where the score can be thought of as a two-dimensional (2D) time-frequency plot, showing schematically that not all frequencies are present at all times. Gabor's proposal was to divide this 2D time-frequency space into cells of area 2p and place one Gaussian per cell. If the Gaussians are considered as a basis set, intuitively a substantial fraction of the Gaussians may be expected to have near-vanishing coefficients.

Figure 1. (a) A section of Beethoven's fifth symphony, showing that if a musical score is viewed as a plot of the time-frequency plane there is strong correlation between frequency and time. Note that most of the time-frequency phase space cells are empty. (b) A schematic representation of the von Neumann lattice in which one Gaussian is placed in every phase space cell of area h. For a color version of this figure, see the color plate section.

It turns out that the identical lattice of Gaussians was discovered by von Neumann 15 years earlier in the context of quantum mechanics, where instead of ? and t the conjugate variables are p and x and the area of the unit cell is h [2]. However, in all respects the formalism is isomorphic. Von Neumann's interest was in a generalized uncertainty principle, but subsequently mathematical physicists explored the properties of the von Neumann lattice as a basis. It was proven that if one Gaussian is placed per cell of area h the von Neumann basis is complete but not overcomplete, provided the width parameter of the Gaussian is appropriate to the cell size [3]. In the late 1970s, Davis and Heller [4] explored the use of the vN basis for solving the time-independent Schrödinger equation (TISE). Their motivation was similar to that of Gabor's. They reasoned that the classical mechanical phase space contour at energy E should provide an excellent guide for where quantum mechanical basis functions are needed. To the extent that basis functions outside the classical contour can be eliminated, the basis should provide a very efficient representation. Some prototypical examples of classical energy contours are illustrated in Fig. 2.

Figure 2. Classical phase space contours for (a) harmonic oscillator Hamiltonian, (b) Coulomb Hamiltonian.

Although the commercial aspects of the representation are probably much larger for audio and image processing than for quantum mechanics, the advantage of the von Neumann representation is potentially much higher in quantum mechanics. The reason is that quantum mechanical calculations for realistic atoms and molecules involve solving a wave equation in 3N degrees of freedom, where N is the number of electrons and nuclei, a dimensionality much higher than one deals with in signal and image processing. Most basis function methods use a tensor product Hilbert space and as a result the number of basis functions grows exponentially with the number of degrees of freedom. This notorious problem is called the "exponential wall" [5]. Although the von Neumann basis functions in multidimensions are direct products of one-dimensional (1D) Gaussians, the Hilbert space after removing the energetically inaccessible Gaussians is not a tensor product Hilbert space. Thus, formally at least, the method has the potential to defeat the exponential wall in basis set calculations.

Due to its intuitive appeal and its potential for simple and efficient representation, the von Neumann representation has attracted interest in the theoretical chemistry community since the late 1970s. Similarly, the Gabor representation has attracted interest in the signal processing community since its invention in 1946, with a peak of interest in the 1980s and 1990s. The development of the theory in these two fields has been nearly independent, with only limited transfer of ideas and methods between these communities. Figure 3a summarizes some of the key milestones in the development of von Neumann/Gabor theory [1, 2, 4, 6-19].

Figure 3. (a) A schematic diagram of the development of the von Neumann/Gabor method in the quantum mechanics and signal processing communities. The development proceeded largely independently. (b) Quotes from the quantum mechanics and signal processing literatures indicating that the von Neumann/Gabor basis on a truncated lattice does not converge.

One of the striking parallels in the development of the method in quantum mechanics and signal processing is that the method never became mainstream in either community. A key reason is undoubtedly the problems encountered in converging the method, problems reported independently in both fields. Figure 3b collects some quotations from the literature, both in quantum mechanics and in signal processing, that testify to the problems with convergence of the method [4, 7, 8, 11, 12].

We have recently discovered a simple but surprising way to converge the von Neumann/Gabor method [15-19]. Our insight was to define a modified von Neumann/Gabor basis in which the boundary conditions are taken to be periodic and band limited. As we show below, this ensures that the representation has exact informational equivalence with the Fast Fourier Transform method, which has been used so profitably for quantum dynamics calculations. In the language of signal processing, the significance of this result is that the modified Gabor basis satisfies a Nyquist-Shannon sampling theorem [20-22], meaning that the representation is exact for functions that are band limited and periodic, and converges exponentially fast for functions that decay exponentially in both time and frequency. The net result is that the periodic von Neumann (pvN) or periodic Gabor (pg) basis combines the best of both worlds: Gaussian flexibility with Fourier accuracy.

One more development is crucial to making the method useful. Although the periodic von Neumann representation has complete informational equivalence with the Fourier representation if the full basis is kept, it turns out that discarding even a single pvN function incurs a considerable error-actually a much larger error than incurred in discarding Fourier functions. In other words, Gabor's original proposal for compression turns out not only to fail, but to have exactly the opposite consequences of what he expected. To understand the problem and its solution, note that the von Neumann basis is non-orthogonal. As a result, the basis functions do not satisfy a Kronecker delta relation ?gm|gn? = dmn, but rather a relation ?bm|gn? = dmn where the {bm} are a set of basis functions biorthogonal to the {gn}. Although the {gn} are localized the {bm} are not. In the implementation of the vN representation as envisioned by Gabor and all subsequent work, the {gn} are the basis functions and therefore the delocalized {bm} determine the coefficients. Our finding was that by interchanging the role of the basis and its biorthogonal basis we obtain a delocalized basis {bm} but the localized functions {gn} now determine the coefficients, many of which are now nearly vanishing.

The remainder of this review is organized as follows. Section II presents the basic theory. Sections III-V present applications, first to femtosecond pulse shaping, then to quantum mechanics (both time independent and time dependent) and finally to audio and image processing. Section VI is a Conclusion with some discussion of future directions.

II. THEORY

A. von Neumann Basis on the Infinite Lattice

The von Neumann basis set [2] is a subset of the "coherent states" of the...