Martin C. Gutzwiller

Periodic Orbits and Classical Quantization Conditions

The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Greens function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability a...

PHASE-INTEGRAL APPROXIMATION IN MOMENTUM SPACE AND THE BOUND STATES OF AN ATOM.

The phase‐integral approximation of the Greens function in momentum space is investigated for a particle of negative energy (bound state) which moves in a spherically symmetric potential. If this potential has a Coulomb‐like singularity at the origin, it is shown that any two momenta can be connected by an infinity of classical trajectories with a fixed energy. The summation of the usual phase and amplitude factors over these trajectories is the approximate Greens function. If there is orbital precession, there are not only poles along the negative energy axis, but also weaker singularities which are not examined in detail. The poles are found at the energies which are given by the semiclassical quantum conditions: angular momentum = (l + ½)ℏ and action integral for the radial motion = (n + ½)2πℏ, where l and n are integers ≥ 0. The residues at these poles give the approximate bound‐state wavefunctions as a product of the asymptotic formula for Legendre polynomials with the asymptotic solution of the ra...

Energy Spectrum According to Classical Mechanics

The phase integral approximation for the Greens function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ‐function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz‐Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spect...

The quantum mechanical Toda lattice, II

Abstract The aim of this paper is to establish the exact quantization conditions for the three-body Toda lattice. The Hamiltonian consists of the kinetic energy for three particles in one dimension, and of the potential energy which couples each particle to its two companions through an exponential spring. After eliminating the center of mass motion, one is left with a system of two degrees of freedom and two constants of motion, the total energy E and a third integral A which commute. Nevertheless, no transformation has been found to separate the classical equations of motion or Schrodingers equation. The wave function is written as a double Laurent series. Its coefficients have to satisfy two sets of recursion relations on a triangular grid where each set insures that we have a simultaneous eigenfunction of E and A. The condition for the convergence of this series can be expressed as the vanishing of a tridiagonal infinite determinant with 1 in the diagonal and the inverse of a third-order polynomial in the first off-diagonals. The coefficients in this polynomial are E and A, and the variable corresponds to a component of the wave vector associated with the wave function. This determinant can be treated exactly as Hills, and yields the 3 components. The condition for the square integrability of the wave function requires the phase angle of the principal minors to be equal to 0, π 3 , or 2π 3 according as the representation of the cyclic groups, for each component of the wave vector. But the third condition follows from the two others. The analogy with the corresponding two-body problem is pointed out.

Bernoulli sequences and trajectories in the anisotropic Kepler problem

The anisotropic Kepler problem is investigated in order to establish the one‐to‐one relation between its trajectories and the binary Bernoulli sequences. The Hamiltonian has a quadratic kinetic energy with an anisotropic mass tensor and a spherically symmetric Coulomb energy. Only trajectories in two dimensions with a negative energy (bound states) are discussed. The previous study of this system was based on extensive numerical computations, but the present work uses only analytical arguments. After a review of the earlier results, their relevance to the understanding of the relation between classical and quantum mechanics is emphasized. The main new result is to show the existence of at least one trajectory corresponding to each binary Bernoulli sequence. The proof employs a number of unusual mathematical tools, although they are all elementary. In particular, the virial as a function of the momenta (rather than the action as a function of the position coordinates) plays a crucial role. Also, different ...

The quantization of a classically ergodic system

Abstract The motion of a charged particle with an anisotropic mass tensor is completely ergodic in two dimensions. Its periodic orbits can be mapped 1-to-1 into the binary sequences of even length. Their action integral is approximated very closely by a quadratic function of the binaries with only one empirical parameter and one scale factor. The trace of the semiclassical Greens function is a sum over all periodic orbits, and can be evaluated by a transformation which Kac used in the discussion of an Ising lattice with long range interaction. The poles of the trace as a function of the energy can be segregated by their discrete symmetry. The real parts agree very well with the quantum mechanical energy levels, while most of the imaginary parts are small, indicating sharp resonances. This is the first calculation of energy levels on the basis of classically ergodic trajectories.

Stochastic behavior in quantum scattering

Abstract A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincare model of hyperbolic geometry, the solutions of Schrodingers equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemanns zeta function on the imaginary axis, at a distance of 1 2 from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimick any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemanns zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned.

The Geometry of Quantum Chaos

The simplest chaotic Hamiltonian systems arise in geometry: a point particle moving freely on a 2 or 3 dimensional manifold of constant negative curvature. There is tremendous variety representing both bound state and scattering situations, quite in contrast to the non-chaotic motion on surfaces of constant positive curvature. The classical mechanics has been studied for almost a century and reduces to an enumeration and description of geodesic lines which in turn are closely connected with the topology of the manifold. Physics enters only in a trivial manner since everything can be scaled with the (kinetic) energy of the particle. In the quantum mechanics of these motions, however, the energy is the essential parameter, and one can ask how the two mechanics are related to each other. Quite surprisingly, Feynmans path integral gives the same answer as its classical approximation, just as in the harmonic oscillator. In 3 dimensions, this result holds for the propagator itself, while in 2 dimensions it is true only for the trace and yields Selbergs trace formula. A few striking examples of constant negative curvature spaces will be shown. Thus, quantum chaos can be understood in terms of classical chaos in spite of their qualitatively different features: fractal for the latter, smooth for the former.

The Transition from Classical to Quantum Mechanics

Since our physical intuition is so firmly grounded in classical mechanics, we have little choice but to advance as far as we can into quantum mechanics along the trails that can be laid out with the help of classical mechanics. To be of help in our context, they have to be usable for regular as well as chaotic dynamical systems, and, therefore, they differ from the ones in most textbooks. The two main guideposts are the classical approximation for the quantum-mechanical propagator in position space and time, as first proposed by Van Vleck in 1928, and the consistent use of the stationary phase method whenever an integral has to be evaluated.

Correlation of Electrons in a Degenerate Band

The ground‐state wavefunction for electrons in a doubly degenerate tight binding band has been investigated by the variational method [M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965). To be referred to as G.III.] The model Hamiltonian consists of a d band with intra‐atomic Coulomb interactions between both d states. The expectation values for the approximate ground state are obtained by counting the various configurations as in the earlier work on correlation in a nondegenerate band. Simple criteria are found for the occurrence of ferromagnetism and of a metal‐insulator transition, using the approach of Brinkman and Rice in the latter case [W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970)]. If the density of states is large at the band edge, the ground state of electrons goes from paramagnetic metallic, through an intermediate ferromagnetic metallic state, to insulating as the intra‐atomic Coulomb interaction is increased.