Abstract
In a demonstration that no physical limitations to computers arise from quantum mechanics and the uncertainty principle, Feynman 1 constructed Hamiltonians for spin systems which performed any desired sequence of logical operations. Although this work showed that there are no intrinsic reasons circuits could not be built at the molecular scale, numerous problems remain in constructing molecular analogs to the standard electronics building blocks: transmission lines, amplifiers, gates and switches. Numerous molecules or molecular systems have been proposed to serve these functions, but such questions such as interconnections, noise isolation, and immunity to thermal and quantum fluctuations remain unanswered. As with the engineering of computers in bulk matter, the engineering of computers in molecular systems requires attention to the detailed dynamics of the system. Part of the problem is at the foundations. Aside from numerical calculations, which are presently beyond the bounds of computer technology, and are likely to remain so for some time hence, the dynamics of large molecules are inaccessible to precise quantitative prediction. On the other hand, an analytical theory of molecular dynamics presents some serious technical challenges and conceptual difficulties. Rapid progress toward a science of information processing at the molecular level awaits the resolution of these fundamental difficulties.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. P. Feynman, “Quantum Mechanical Computers,” Foundations of Physics 16: 507 (1986).
R. R. Birge, “The Photophysics of Light Transduction in Rhodopsin and Bacteriorhodopsin,” Ann. Rev. Biophys. Bioeng., 10: 315 (1981).
B. W. Kobilka, T. S. Kobilka, K. Daniel, J. W. Regan, M. G. Caron and R. J. Lcfkowüz Science, 240: 1310 (1988).
R. R. Birge, L. A. Findsen and B. M. Pierce, “Molecular dynamics of the Primary Photochemical Event in Bacteriorhodopsin: Theoretical Evidence for an Excited State Assignment for the J Intermediate,” J. Am. Chem. Soc., 109: 5041 (1987).
S. Pnevmatikos, “Soliton Dynamics of Hydrogen-Bonded Networks: A Mechanism for Proton Conductivity,” Phys. Rev. Lett., 60: 1534 (1988).
A. J. Legett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, “Dynamics of the Dissipative Two-State System,” Rev. Mod. Phys., 59: 1 (1987).
C. DeWitt-Morette, A. Maheshwari, and B. Nelson, “Path Integration in Non- Relativistic Quantum Mechanics,” Physics Reports, 50: 255 (1979).
B. Simon, “Semiclassical Analysis of Low Lying Eigenvalues I. Non-Degenerate Minima: Asymptotic Expansions,” Ann. Inst. Henri Poincare, A38: 295 (1983).
B. Simon, “Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling,” Annals of Mathematics, 120: 89 (1984).
J. N. L. Connor, “Catastrophes and Molecular Collisions.” Molecular Physics, 31: 33 (1976).
B. Simon, “Large Orders and Summability of Eigenvalue Perturbation Theory: A Mathematical Overview,” International journal of Quantum Chemistry, 21: 3 (1982).
V. I. Arnold, “Remarks on the Stationary Phase Method and Coexter Numbers,” Uspekhi Mat. Nauk., 28: 17 (1973).
J. J. Duistermaat, “Oscillatory Integrals, Lagrange Immersions and Unfolding of Singularities,” Communications on Pure and Applied Mathematics, 27: 207 (1974).
A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin (1986)
H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer-Verlag, New York. (1987).
R. R. Birge and A. F. Lawrence, “Optical Random Access Memory Based on Bacteriorhodopsin,” These Proceedings.
W. H. Miller and T. F. George, “Semiclassical Theory of Electronic Transitions in Low Energy Atomic and Molecular Collisions Involving Several Nuclear Degrees of Freedom,” J. Chem. Phys., 56: 5637 (1972).
I. M. Sigal and A. Soffer, “The N-Particle Scattering Problem: Asymptotic Completeness for Short-Range Systems,” Annals of Mathematics, 126: 35(1987).
G. A. Hagedorn, “High Order Corrections to the Time-Dependent Bom-Oppenheimer Approximation I: Smooth Potentials,” Annals of Mathematics, 124: 571 (1986).
R. Bott, “Lectures on Morse Theory, Old and New,” Bulletin American Math. Soc., 7:2 331 (1982).
J. B. Keller and D. W. McLaughlin, “The Feynman Integral,” American Math. Monthly, 82: 457 (1975).
L. S. Schulman, Techniques and Applications of Path Integration, John Wiley and Sons, New York. (1981).
S. Albeverio and R. Höegh-Krohn, “Oscillatory Integrals and the Method of Stationary Phase in Infinitely Many Dimensions, with Applications to the Classical Limit of Quantum Mechanics I.,” Inventiones Math., 40: 59 (1977).
E. Merzbacher, Quantum Mechanics, (2nd Edition), John Wiley & Sons, Inc., New York (1970).
B. C. Eu, “Quantum Theory of Large Amplitude Vibrational Motions in a One- Dimensional Morse Chain,” J. Chem. Phys., 73: 2405 (1980).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Plenum Press, New York
About this chapter
Cite this chapter
Lawrence, A.F., Birge, R.R. (1989). Mathematical Problems Arising in Molecular Electronics: Global Geometry and Dynamics of the Double-Well Potential. In: Hong, F.T. (eds) Molecular Electronics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-7482-8_42
Download citation
DOI: https://doi.org/10.1007/978-1-4615-7482-8_42
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4615-7484-2
Online ISBN: 978-1-4615-7482-8
eBook Packages: Springer Book Archive