Journal of Mathematical Physics
Feedback | Help | Exit
Search:
   
 
 
 
Previous Article
Exact Quantization Conditions. II
By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that...
Next Article
Mth Power of an N × N Matrix and Its Connection with the Generalized Lucas Polynomials
The Mth power of an N × N matrix is expressed via the Cayley-Hamilton theorem as a linear combination of the lower powers of the matrix. The polynomial coefficients of the lower powers of the ma...

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

J. Math. Phys. 10, 1458 (1969); doi:10.1063/1.1664991

Issue Date: August 1969

You are logged in to this journal.

H. R. Lewis, Jr. and W. B. Riesenfeld
Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico
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. ©1969 The American Institute of Physics
History: Received 8 July 1968
Permalink: http://link.aip.org/link/?JMAPAQ/10/1458/1
FULL TEXT OPTIONS   (FREE)
Download PDF (1084 kB) View Cart

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (10)
View in Separate Window/Tab
  1. H. R. Lewis, Jr., J. Math. Phys. 9, 1976 (1968); [ISI]
    2. also, H. R. Lewis, Jr., Phys. Rev. Letters 18, 510, 636 (1967).
  2. The two special systems that we consider in Secs. III and IV have been treated along different lines by M. Kolsrud: (a) “Exact Quantum Dynamical Solutions for Oscillator-Like Systems,” Institute for Theoretical Physics, University of Oslo (Norway), Institute Report No. 28 (1965);
    3. (b) Kgl. Norske Videnskab. Selskabs Forh. 31, No. 5 (1958).
    (c) Phys. Rev. 104, 1186 (1956). [ISI]
  3. L. Landau, Z. Physik 64, 629 (1930).
  4. R. B. Dingle, Proc. Roy. Soc. (London) A211, 500 (1952).
  5. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1965), 2nd. ed., p. 426. There is an error in this derivation: the wavefunction is assumed proportional to eimphi, but the subsequent formulas are derived for e−imphi.
  6. A method for expressing the general solution of Eq. (45) for arbitrary Omega(t) in terms of independent solutions of the equations for a classical oscillator has been described in Ref. 1.
  7. P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1947), 3rd ed.
    8. Also see A. Messiah, Quantum Mechanics (Interscience Publishers, New York, 1962), Vol. I.
  8. For the present we are omitting the time label t in our notation for these eigenstates. When it is required for clarity, we shall replace |s> by |s;t> to denote an eigenstate at time t.
  9. This derivation is closely related to a derivation of an analogous invariant for the corresponding classical system. The treatment of the classical system is given in H. R. Lewis, Jr., Phys. Rev. 172, 1313 (1968). [ISI]
  10. L. Infeld, Phys. Rev. 59, 737 (1941). [ISI]
    11. Later developments of this interesting method can be found in the following references: (a) T. Inui, Progr. Theoret. Phys. (Kyoto) 3, 168, 244 (1948);
    (b) L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 (1951);
    (c) A. Joseph, ibid. 39, 829 (1967); [ISI]
    (d) C. A. Coulson and A. Joseph, ibid. 39, 838 (1967). [ISI]

Copyright © American Institute of Physics