Abstract
In classical mechanics, every function F on phase space generates a one-parameter group of diffeomorphisms exp(t LX F .) (t ∈ R and LX F is the Lie derivative with respect to the Hamiltonian vector field corresponding to F). Similarly, we learned in §2.4 that in quantum theory every observable a is associated with a one-parameter group of automorphisms b → exp(iat)b exp(—iat). One of the basic postulates quantum theory is that, in units with ħ = 1, the groups generated by the Cartesian position and momentum coordinates x i and p j of n particles (j = 1, ....,n) are the same as classically, i.e., displacements respectively in the momenta and positions. Since x i and p j do not have bounded spectra, and hence cannot be represented by bounded operators, it is convenient to consider instead the bounded functions\(\exp \left( {i\sum\limits_{i = 1}^n {{X_j} \cdot {S_j}} } \right)and\exp \left( {i\sum\limits_{i = 1}^n {{p_j} \cdot {r_j}} } \right),{S_j}{r_j} \in {R^3}\) so as not to have domain questions to worry about. The group of automorphisms can be written in terms of them as follows:
Phase space is the arena of classical mechanics. The algebra of observables in quantum mechanics is likewise constructed with position and momentum, so this section covers the properties of those operators.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Cohen-Tannoudji, B. Diu, and F. Laloë: Quantum Mechanics. New York, Wiley, 1979.
A. Galindo and P. Pascual: Mecânica Cudntica. Madrid, Alhambra, 1978.
G. Grawert. Quantenmechanik. Wiesbaden, Akademie Verlagsgesellschaft, 1977.
R. Jost: Quantenmechanik, in two volumes. Zurich, Verlag des Vereins der Mathematiker und Physiker an der ETH, 1969.
A. Messiah: Quantum Mechanics, in two volumes. Amsterdam, North-Holland, 1961–1962.
F.L. Pilar: Elementary Quantum Chemistry. New York, McGraw-Hill, 1968. L.I. Schiff: Quantum Mechanics. New York, McGraw-Hill, 1968.
B. Simon: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton, Princeton University Press, 1974.
H. Weyl: Gruppentheorie und Quantenmechanik. Leipzig, S. Hirzel, 1931.
A.M. Perelomov: Coherent States for Arbitrary Lie Groups. Commun. Math. Phys. 26, 222–236, 1972.
R. Jost: The General Theory of Quantized Fields. Providence, American Mathematical Society, 1965.
R.F. Streater and A.S. Wightman: PCT, Spin, Statistics, and All That. New York, Benjamin, 1964.
H. Weyl, op. cit.
A.R. Edmonds: Angular Momentum in Quantum Mechanics. Princeton, Princeton University Press, 1974.
P. Ehrenfest: Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z. Phys. 45, 455–457, 1927.
T. Kato: On th Eigenfunctions of Many-Particle Systems in Quantum Mechanics. Commun. on Pure and Appl. Math. 10, 151–177, 1957.
K. Hepp: The Classical Limit for Quantum Mechanical Correlation Functions. Commun. Math. Phys. 35, 265–277, 1974.
W.O. Amrein, Ph.A. Martin, and B. Misra: On the Asymptotic Condition of Scattering Theory. Helv. Phys. Acta 43, 313–344, 1970.
T. Kato: Wave Operators and Similarity for some Non-Selfadjoint Operators. Math. Ann. 162, 258–279, 1966.
S. Agmon: Spectral Properties of Schrödingher Operators and Scattering Theory. Ann. Scuola Norm. Sup. Pisa, Cl. di Sci., ser. IV, 2, 151–218, 1975.
P. Deift and B. Simon: A Time-Dependent Approach to the Completeness of Multiparticle Quantum Systems. Commun. on Pure and Appl. Math. 30, 573–583, 1977.
A. Weinstein and W. Stenger: Methods of Intermediate Problems for Eigenvalues: Theory and Ramifications. New York, Academic Press, 1972.
M.F. Barnsley: Lower Bounds for Quantum Mechanical Energy Levels. J. Phys. A11, 55–68, 1978.
B. Baumgartner: A Class of Lower Bounds for Hamiltonian Operators. J. Phys. A12, 459467, 1979.
R.J. Duffin: Lower Bounds for Eigenvalues. Phys. Rev. 71, 827–828, 1947.
H. Grosse, private communication.
P. Hertel, H. Grosse, and W. Thirring: Lower Bounds to the Energy Levels of Atomic and Molecular Systems. Acta Phys. Austr. 49, 89–112, 1978.
B. Simon: An Introduction to the Self-Adjointness and Spectral Analysis of Schrödinger Operators. In: The Schrödinger Equation, op. cit.,p. 19.
V. Glaser, H. Grosse, A. Martin, and W. Thirring: A Family of Optimal Conditions for the Absence of Bound States in a Potential. In: Studies in Mathematical Physics, op. cit.,p.169.
W.O. Amrein, J.M. Jauch, and K.B. Sinha: Scattering Theory in Quantum Mechanics: Physical Principles and Mathematical Methods. Lecture Notes and Supplements in Physics, vol. 16. New York, Benjamin, 1977.
M.L. Goldberger and K.M. Watson: Collision Theory. New York, Wiley, 1964.
R.G. Newton: Scattering Theory of Waves and Particles. New York, McGraw-Hill, 1966.
W. Sandhas: The N-Body Problem. Acta Phys. Austr. Suppl., Vol. 13, Vienna and New York, Springer, 1974.
J.R. Taylor: Scattering Theory. New York, Wiley, 1972
K. Osterwalder, ed: Mathematical Problems in Theoretical Physics. Proc. Int. Conf. on Math. Phys. (Lausanne, Switzerland, Aug. 1979 ). Berlin-Heidelberg-New York, Springer, 1980.
M.J. Englefield: Group Theory and the Coulomb Problem. New York, Interscience, 1972.
H. Grosse, H.-R. Grümm, H. Narnhofer, and W. Thirring: Algebraic Theory of Coulomb Scattering. Acta Phys. Austr. 40, 97–103, 1974.
E. Nelson: Time-Ordered Operator Products of Sharp-Time Quadratic Forms. J. Func. Anal. 11, 211–219, 1972.
W. Faris and R. Lavine: Commutators and Self-Adjointness of Hamiltonian Operators. Commun Math. Phys. 35, 39–48, 1974.
R. Lavine: Spectral Densities and Sojourn Times. In: Atomic Scattering Theory, J. Nuttall, ed. London, Ontario, University of Western Ontario Press, 1978.
R. Lavine and M. O’Carroll: Ground State Properties and Lower Bounds for Energy Levels of a Particle in a Uniform Magnetic Field and External Potential. J. Math. Phys. 18, 1908–1912, 1977.
H. Narnhofer and W. Thirring: Convexity Properties for Coulomb Systems. Acta Phys. Austr. 41, 281–297, 1975.
T. Kinoshita: Ground State of the Helium Atom. Phys. Rev. 105, 1490–1502, 1957.
C.L. Pekeris: Ground State of Two-Electron Atoms. Phys. Rev. 112, 1649–1658, 1958.
K. Frankowski and C.L. Pekeris: Logarithmic Terms in the Wave Function of the Ground State of Two-Electron Atoms.Phys. Rev. 146, 46–49, 1966.
R. Ahlrichs, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and J.D. Morgan: Bounds on the Decay of Electron Densities with Screening. Phys. Rev. A23, 2106, 1981.
S. Agmon: Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators. Mathematical Notes, vol. 29. Tokyo, University of Tokyo Press, 1982.
S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.O. Sorensen: The electron density is smooth away from the nuclei. Commun. Math. Phys. 228, 401–415, 2002.
W. Faris: Inequalities and Uncertainty Principles. J. Math. Phys. 19, 461–466, 1978.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Thirring, W. (2002). Quantum Dynamics. In: Quantum Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05008-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-05008-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07711-1
Online ISBN: 978-3-662-05008-8
eBook Packages: Springer Book Archive