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American Journal of Physics

Invariant operators for quadratic Hamiltonians

Mark Andrews

Published Apr 1, 1999• 23 citations
Publisher

Abstract

The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.

DOI
10.1119/1.19259

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The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2(Rn)L^2(\mathbb R^n)L2(Rn) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.
Using the time-dependent annihilation and creation operators, the invariant operators, for a free mass and an oscillator, we find the coherent-squeezed state representation of a travelling general Gaussian wave packet with initial expectation values, x0x_0x0​ and p0p_0p0​, of the position and momentum and variances, Δx0Δx_0Δx0​ and Δp0Δp_0Δp0​. The initial general Gaussian wave packet takes, up to a normalization factor, the form eip0x/ℏe−(1∓iδ)(x−x0)2/4(Δx0)2e^{i p_0 x/\hbar} e^{- (1 \mp i δ) (x - x_0)^2 / 4 (Δx_0)^2}eip0​x/ℏe−(1∓iδ)(x−x0​)2/4(Δx0​)2, where δ=(2Δx0Δp0/ℏ)2−1δ= \sqrt{(2Δx_0 Δp_0/\hbar)^2 -1}δ=(2Δx0​Δp0​/ℏ)2−1​ denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation δ=2Δ(xp)0/ℏδ= 2Δ(xp)_0 / \hbarδ=2Δ(xp)0​/ℏ. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form eipcx/ℏe−(1−2iΔ(xp)t/ℏ)(x−xc)2/4(Δxt)2e^{i p_c x/\hbar} e^{- (1 - 2 i Δ(xp)_t/\hbar) (x - x_c)^2 / 4 (Δx_t)^2}eipc​x/ℏe−(1−2iΔ(xp)t​/ℏ)(x−xc​)2/4(Δxt​)2 and the centroid follows the the classical trajectory with xc(t)x_c(t)xc​(t) and pc(t)p_c(t)pc​(t). The position variance is found to have additionally a linearly time-dependent term proportional to δδδ with both positive and negative signs.
The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined via the solution of a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, tunnelling currents, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied via linearization of the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. Furthermore, these two solutions also provide the time-dependent Green function for the systems propagation for any given initial condition, not only for Gaussians. It is also possible to obtain the corresponding Wigner function and generalized creation and annihilation operators in this way. In searching for problems with analytical solutions, the crucial point is to identify systems that have either a wave packet solution with time-dependent width (in closed form) that obeys a Riccati equation, or a classical parametric oscillator with frequency ω(t) that still enables an analytical solution of the Newtonian equation. Comparing the wave packets with the solution of the diffusion equation – also a Gaussian with time-dependent width – leads to a real Riccati equation. A first attempt at "complexifying" this equation turns out to be problematic. Nevertheless, it provides a clue to time-dependent frequencies inversely proportional to time that lead to Newtonian equations with analytical solutions. Following a detailed analysis of the corresponding quantum dynamics and energetics, possible extensions of the method are indicated.
\n\nUsing the ABCD formalism of atom optics, the generating function method and the invariant operator method (Liouville-von Neumann picture), I show in this letter how the quantum time evolution of any wave packet can be entirely determined by a natural symplectic invariant, which provides both the (non-Hermitian) invariant operator and the adequate eigenvalue. Within this framework, the quantum propagator does not appear any more as the fundamental kernel of integration but merely as a particular property of this invariant operator. The symplectic invariant method is interpreted in terms of creation/annihilation operators of N-dimensional Hermite-Gaussian modes.\n
We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For simplicity, we consider a free particle with a variable mass M(t), a particle with a variable mass M(t) in an alternating homogeneous field, and a harmonic oscillator with a variable mass M(t) and frequency ω(t) subject to a variable force F(t). To construct the evolution operators for these systems in an explicit disentangled form, we use a simple technique to find the general solution of a certain class of differential and finite-difference nonstationary Schrödinger-type equations of motion and also the operator identities of the Baker–Campbell–Hausdorff type. With known evolution operators, we can easily find the most general form of the propagators, invariants of any order, and wave functions and establish a unitary relation between systems. Results known in the literature follow from the obtained general results as particular cases.
We investigate how wave functions evolve with time in the harmonic oscillator. We first review the periodicity properties over each multiple of a quarter of the classical oscillation period. Then we show that any wave function can be simply transformed so that its centroid, defined by the expectation values of position and momentum, remains at rest at the center of the oscillator. This implies that we need only consider the evolution of this restricted class of wave functions; the evolution of all others can be reduced to these. The evolution of the spread in position Δx and momentum Δp throws light on energy and uncertainty and on squeezed and coherent states. Finally, we show that any wave function can be transformed so that Δx and Δp do not change with time and that the evolution of all wave functions can easily be found from the evolution of those at rest at the origin with unchanging Δx and Δp.