The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
A class of exact invariants for oscillator systems whose Hamiltonians areH=(1/2ε)[p2+Ω2(t)q2]is given in closed form in terms of a function ρ(t) which satisfies ε2d2ρ/dt2+Ω2(t)ρ−ρ−3=0. Each particular solution of the equation for ρ determines an invariant. The invariants are derived by applying an asymptotic theory due to Kruskal to the oscillator system in closed form. As a consequence, the results are more general than the asymptotic treatment, and are even applicable with complex Ω(t) and quantum systems. A generating function is given for a classical canonical transformation to a class of new canonical variables which are so chosen that the new momentum is any particular member of the class of invariants. The new coordinate is, of course, a cyclic variable. The meaning of the invariants is discussed, and the general solution for ρ(t) is given in terms of linearly independent solutions of the equations of motion for the classical oscillator. The general solution for ρ(t) is evaluated for some special cases. Finally, some aspects of the application of the invariants to quantum systems are discussed.
Time-independent description of rapidly driven systems in the presence of friction: Multiple scale perturbation approach Chaos 22, 013131 (2012) Limitations on cloning in classical mechanics J. Math. Phys. 53, 012902 (2012) Equivalence problem for the orthogonal webs on the 3-sphere J. Math. Phys. 52, 053509 (2011) A Foucault's pendulum design Rev. For a classical Hamiltonian H = (1/2) p2 + V(q,t) with an arbitrary time-dependent potential V(q,t ),exactinvariantsthatcanbeexpressedasseriesinpositivepowersof p, I (q,p,t ) = ~: = opYn (q,t), are examined. The method is based on direct use of the equation dIldt = aIlat + [I,H] = O. A recursion relation for the coefficients! n (q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in pare determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.
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