time-dependent harmonic oscillators

1 Time-Dependent Harmonic Oscillators

Nonlinear differential equations arise naturally in physics. Riccati equations and Ermakov systems enter quantum theory in the study of exact Gaussian wave-packet solutions of the time-dependent Schrödinger equation, including the time-dependent harmonic oscillator and free-particle cases.

One of the simplest nonlinear equations of this type is the Milne–Pinney equation,

2 k-- x¨+ ω (t)x = x3,

where k is a real constant. Equivalently,

x¨= − ω2(t)x + k-. x3

1.1 Ermakov Systems

The equation was introduced by V. P. Ermakov in the nineteenth century in the study of first integrals for the time-dependent harmonic oscillator. Lie had previously characterized classes of non-autonomous first-order systems that admit a superposition rule,

dxi= Yi(t,x ), i = 1,...,n. dt

A classical Ermakov system consists of a time-dependent harmonic oscillator coupled to a Milne–Pinney equation with the same frequency:

¨η + ω2 (t)η = 0,

2 k ¨α + ω (t)α = -3. α

Such systems appear in studies of time-dependent oscillators, wave-packet dynamics, Bose–Einstein condensates, and cosmological models.

For the system above, an Ermakov–Lewis invariant is

1 [ ( η )2] IL = -- (α ˙η − η ˙α)2 + k -- . 2 α

A direct differentiation, followed by use of the two equations of motion, gives

dIL-= 0. dt

In quantum-mechanical applications, α(t) is often interpreted as an auxiliary amplitude related to the width of a Gaussian wave packet, while η(t) may describe the corresponding classical oscillator trajectory. Further insight into relations of this type can also be obtained through the Riccati equation.

References

[1] Dieter Schuch. Riccati and Ermakov Equations in Time–Dependent and Time–Independent Quantum Systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 4, 043, 16 pages (2008).

[2] R. Goodall and P. G. L. Leach. Generalised Symmetries and the Ermakov–Lewis Invariant. Journal of Nonlinear Mathematical Physics, 12(1), 15–26 (2005).

[3] B. Grammaticos and B. Dorizzi. Two-dimensional time-dependent Hamiltonian systems with an exact invariant. Journal of Mathematical Physics 25, 2194–2199 (1984).

[4] R. S. Kaushal. Quantum analogue of Ermakov systems and the phase of the quantum wave function. International Journal of Theoretical Physics 40, 835–847 (2001).

[5] H. J. Korsch and H. Laurent. Milne’s differential equation and numerical solutions of the Schrödinger equation. I. Bound-state energies for single- and double-minimum potentials. J. Phys. B: At. Mol. Phys. 14, 4213–4230 (1981).

[6] H. J. Korsch, H. Laurent, and S. Möhlenkamp. Milne’s differential equation and numerical solutions of the Schrödinger equation. II. Complex energy resonance states. J. Phys. B: At. Mol. Phys. 15, 1–15 (1982).

[7] D. Schuch. Relations between wave and particle aspects for motion in a magnetic field, in New Challenges in Computational Quantum Chemistry, eds. R. Broer, P. J. C. Aerts, and P. S. Bagus, University of Groningen, 255–269 (1994).

[8] M. Maamache, A. Bounames, and N. Ferkous. Comment on “Wave function of a time-dependent harmonic oscillator in a static magnetic field.” Phys. Rev. A 73, 016101 (2006).

[9] J. R. Ray. Time-dependent invariants with applications in physics. Lett. Nuovo Cim. 27, 424–428 (1980).

[10] W. Sarlet. Class of Hamiltonians with one degree of freedom allowing applications of Kruskal’s asymptotic theory in closed form. II. Ann. Phys. (N.Y.) 92, 248–261 (1975).

[11] H. R. Lewis and P. G. L. Leach. Exact invariants for a class of time-dependent nonlinear Hamiltonian systems. J. Math. Phys. 23, 165–175 (1982).

[12] P. G. L. Leach and A. Andriopoulos. The Ermakov Equation: A Commentary. Applicable Analysis and Discrete Mathematics 2, 146–157 (2008).

[13] P. G. L. Leach, A. Karasu, M. C. Nucci, and A. Andriopoulos. Ermakov’s Superintegrable Toy and Non-Local Symmetries. SIGMA 1, 018 (2005).

[14] M. Sebawe Abdalla and P. G. L. Leach. Linear and quadratic invariants for the transformed Tavis–Cummings model. J. Phys. A: Math. Gen. 36, 12205–12221 (2003).

[15] M. Sebawe Abdalla and P. G. L. Leach. Wigner functions for time-dependent coupled linear oscillators via linear and quadratic invariant processes. J. Phys. A: Math. Gen. 38, 881–893 (2005).

[16] R. S. Kaushal. Classical and Quantum Mechanics of Noncentral Potentials: A Survey of 2D Systems. Springer, Heidelberg (1998).

[17] V. Ermakov. Second-order differential equations. Conditions of complete integrability. Universita Izvestia Kiev, Series III, 9, 1–25 (1880); translation by A. O. Harin.

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