DOI: 10.1063/1.1726550. On the Exponential Form of Time-Displacement Operators in Quantum Mechanics

On the Exponential Form of Time-Displacement Operators in Quantum Mechanics

10.1063/1.1726550

Crossref journal-article

AIP Publishing

The Journal of Chemical Physics (317)

Abstract

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.

Bibliography

Pechukas, P., & Light, J. C. (1966). On the Exponential Form of Time-Displacement Operators in Quantum Mechanics. The Journal of Chemical Physics, 44(10), 3897â3912.

Authors 2

Philip Pechukas (first)
John C. Light (additional)

References 27 Referenced 511

{'key': '2024020900580504300_r1a', 'first-page': '43', 'volume': '25', 'year': '1963', 'journal-title': 'Progr. Theoret. Phys. (Kyoto) Suppl.'} / Progr. Theoret. Phys. (Kyoto) Suppl. (1963)
10.1063/1.1704297 / J. Math. Phys. (1965)
10.1103/PhysRev.75.486 / Phys. Rev. (1949)
10.1103/PhysRev.84.108 / Phys. Rev. (1951)
10.1002/cpa.3160070404 / Commun. Pure Appl. Math. (1954)
{'key': '2024020900580504300_r5'}
{'key': '2024020900580504300_r6'}
{'key': '2024020900580504300_r7', 'first-page': '140', 'volume': '36', 'year': '1963', 'journal-title': 'Helv. Phys. Acta'} / Helv. Phys. Acta (1963)
{'key': '2024020900580504300_r8'}
{'key': '2024020900580504300_r9'}
{'key': '2024020900580504300_r10'}
10.1103/RevModPhys.20.367 / Rev. Mod. Phys. (1948)
10.1017/S030500410002524X / Proc. Cambridge Phil. Soc. (1949)
10.1139/p58-038 / Can. J. Phys. (1958)
10.1063/1.1696777 / J. Chem. Phys. (1965)
{'key': '2024020900580504300_r14'}
{'key': '2024020900580504300_r15'}
10.1103/RevModPhys.20.367 / Rev. Mod. Phys. (1948)
10.1103/PhysRev.81.848 / Phys. Rev. (1951)
{'key': '2024020900580504300_r17'}
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{'key': '2024020900580504300_r20'}
{'key': '2024020900580504300_r21'}
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Dates

Type	When
Created	20 years, 7 months ago (Jan. 9, 2005, 3:17 a.m.)
Deposited	1 year, 6 months ago (Feb. 8, 2024, 7:58 p.m.)
Indexed	2 months, 3 weeks ago (June 10, 2025, 2:44 p.m.)
Issued	59 years, 3 months ago (May 15, 1966)
Published	59 years, 3 months ago (May 15, 1966)
Published Print	59 years, 3 months ago (May 15, 1966)

Funders 0

None

BibTeX

@article{Pechukas_1966, title={On the Exponential Form of Time-Displacement Operators in Quantum Mechanics}, volume={44}, ISSN={1089-7690}, url={http://dx.doi.org/10.1063/1.1726550}, DOI={10.1063/1.1726550}, number={10}, journal={The Journal of Chemical Physics}, publisher={AIP Publishing}, author={Pechukas, Philip and Light, John C.}, year={1966}, month=may, pages={3897–3912} }

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