An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field

Journal of Mathematical Physics - Tập 10 Số 8 - Trang 1458-1473 - 1969
Harry R. Lewis1, W. B. Riesenfeld1
1Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico

Tóm tắt

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.

Từ khóa


Tài liệu tham khảo

1968, J. Math. Phys., 9, 1976, 10.1063/1.1664532

1967, Phys. Rev. Letters, 18, 510, 10.1103/PhysRevLett.18.510

1958, Kgl. Norske Videnskab. Selskabs Forh., 31

1956, Phys. Rev., 104, 1186, 10.1103/PhysRev.104.1186

1930, Z. Physik, 64, 629, 10.1007/BF01397213

1952, Proc. Roy. Soc. (London), A211, 500

1968, Phys. Rev., 172, 1313, 10.1103/PhysRev.172.1313

1941, Phys. Rev., 59, 737, 10.1103/PhysRev.59.737

1948, Progr. Theoret. Phys. (Kyoto), 3, 168, 10.1143/ptp/3.2.168

1951, Rev. Mod. Phys., 23, 21, 10.1103/RevModPhys.23.21

1967, Rev. Mod. Phys., 39, 829, 10.1103/RevModPhys.39.829

1967, Rev. Mod. Phys., 39, 838, 10.1103/RevModPhys.39.838

Thông tin
Thông tin xuất bản

Journal of Mathematical Physics

Tập 10 Số 8

1458-1473

Thông tin tác giả