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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Razavy, Mohsen. Quantum theory of tunneling. Singapore : World Scientific Publishing, ©2014. xxiv, 767 pages |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Mohsen Razavy |

ISBN: | 9789814525022 9814525022 |

OCLC Number: | 878144511 |

Description: | 1 online resource (792 pages) : illustrations |

Contents: | 1. A brief history of quantum tunneling -- 2. Some basic questions concerning quantum tunneling. 2.1. Tunneling and the uncertainty principle. 2.2. Asymptotic form of decay after a very long time. 2.3. Initial stages of decay. 2.4. Solvable models exhibiting different stages of decay -- 3. Simple solvable problems. 3.1. Confining double-well potentials. 3.2. Tunneling through barriers of finite extent. 3.3. Tunneling through a series of identical rectangular barriers. 3.4. Eckart's potential. 3.5. Double-well morse potential. 3.6. A solvable asymmetric double-well potential -- 4. Time-dependence of the wave function in one-dimensional tunneling. 4.1. Time-dependent tunneling for a [symbol]-function barrier. 4.2. An asymptotic expression in time for the transmission of a wave packet -- 5. Semiclassical approximations. 5.1. The WKB approximation. 5.2. Method of Miller and good. 5.3. Calculation of the splitting of levels in a symmetric double-well potential using WKB approximation. 5.4. Energy eigenvalues for motion in a series of identical barriers. 5.5. Tunneling in momentum space. 5.6. The Bremmer series -- 6. Generalization of the Bohr-Sommerfeld quantization rule and its application to quantum tunneling. 6.1. The Bohr-Sommerfeld method for tunneling in symmetric and asymmetric wells. 6.2. Numerical examples -- 7. Gamow's theory, complex eigenvalues, and the wave function of a decaying state. 7.1. Solution of the Schrodinger equation with radiating boundary condition. 7.2. Green's function for the time-dependent Schrodinger equation with radiating boundary conditions. 7.3. The time development of a wave packet trapped behind a barrier. 7.4. Method of auxiliary potential. 7.5. Determination of the wave function of a decaying state. 7.6. Some instances where WKB approximation and the Gamow formula do not work -- 8. Tunneling in symmetric and asymmetric local potentials and tunneling in nonlocal and quasi-solvable barriers. 8.1. Tunneling in double-well potentials. 8.2. Tunneling when the barrier is nonlocal. 8.3. Tunneling in separable potentials. 8.4. Quasi-solvable examples of symmetric and asymmetric double-wells. 8.5. Gel'fand-Levitan method. 8.6. Darboux's method. 8.7. Optical potential barrier separating two symmetric or asymmetric wells -- 9. Classical descriptions of quantum tunneling. 9.1. Coupling of a particle to a system with infinite degrees of freedom. 9.2. Classical trajectories with complex energies and quantum tunneling -- 10. Tunneling in time-dependent barriers. 10.1. Multi-channel Schrodinger equation for periodic potentials. 10.2. Tunneling through an oscillating potential barrier. 10.3. Separable tunneling problems with time-dependent barriers. 10.4. Penetration of a particle inside a time-dependent potential barrier -- 11. Decay width and the scattering theory. 11.1. One-dimensional scattering theory and escape from a potential well. 11.2. Scattering theory and the time-dependent Schrodinger equation. 11.3. An approximate method of calculating the decay widths. 11.4. Time-dependent perturbation theory applied to the calculation of decay widths of unstable states. 11.5. Early stages of decay via tunneling. 11.6. An alternative way of calculating the decay width using the second order perturbation theory. 11.7. Tunneling through two barriers. 11.8. R-matrix formulation of tunneling problems. 11.9. Decay of the initial state and the Jost function -- 12. The method of variable reflection amplitude applied to solve multichannel tunneling problems. 12.1. Mathematical formulation. 12.2. Variable partial wave phase method for central potentials. 12.3. Matrix equations and semi-classical approximation for many-channel problems -- 13. Path integral and its semiclassical approximation in quantum tunneling. 13.1. Application to the S-wave tunneling of a particle through a central barrier. 13.2. Method of euclidean path integral. 13.3. Other applications of the path integral method in tunneling. 13.4. Complex time, path integrals and quantum tunneling. 13.5. Path integral and the Hamilton-Jacobi coordinates. 13.6. Path integral approach to tunneling in nonlocal barriers. 13.7. Remarks about the semiclassical propagator and tunneling problem -- 14. Heisenberg's equations of motion for tunneling. 14.1. The Heisenberg equations of motion for tunneling in symmetric and asymmetric double-wells. 14.2. Heisenberg's equations for tunneling in a symmetric double-well. 14.3. Heisenberg's equations for tunneling in an asymmetric double-well. 14.4. Tunneling in a potential which is the sum of inverse powers of the radial distance. 14.5. Klein's method for the calculation of the eigenvalues of a confining double-well potential. 14.6. Finite difference method for tunneling in confining potentials. 14.7. Finite difference method for one-dimensional tunneling. 15. Wigner distribution function in quantum tunneling. 15.1. Wigner distribution function and quantum tunneling. 15.2. Wigner trajectory for tunneling in phase space. 15.3. Entangled classical trajectories. 15.4. Wigner distribution function for an asymmetric double-well. 15.5. Wigner trajectory for an oscillating wave packet. 15.6 Margenau-Hill distribution function for a double-well potential -- 16. Decay widths of Siegert states, complex scaling and dilatation transformation. 16.1. Siegert resonant states. 16.2. A numerical method of determining Siegert resonances. 16.3. Riccati-Pade method of calculating complex eigenvalues. 16.4. Complex rotation or scaling method. 16.5. Milne's method. 16.6. Complex energy resonance states calculated by Milne's differential equation. 16.7. S-wave scattering from a delta function potential. 16.8. Resonant states for solvable potentials -- 17. Multidimensional quantum tunneling. 17.1. The semiclassical approach of Kapur and Peierls. 17.2. Wave function for the lowest energy state. 17.3. Calculation of the low-lying wave functions by quadrature. 17.4. Semiclassical wave function. 17.5. Tunneling of a Gaussian wave packet. 17.6. Interference of waves under the barrier. 17.7. Penetration through two-dimensional barriers. 17.8. Method of quasilinearization applied to the problem of multidimensional tunneling. 17.9. Solution of the general two-dimensional problems. 17.10. The most probable escape path. 17.11. An extension of the Hamilton-Jacobi theory and its application for solving multidimensional tunneling problems. 17.12. A time-dependent approach to the problem of tunneling in two dimensions -- 18. Group and signal velocities. 18.1. Exact solution of the problem of tunneling in a constant barrier -- 19. Time-delay, reflection time operator and minimum tunneling time. 19.1. Time-delay caused by tunneling. 19.2. Time-delay for tunneling of a wave packet. 19.3. Landauer and Martin criticism of the definition of the time-delay in quantum tunneling. 19.4. Other approaches to the tunneling time problem. 19.5. Time-delay in multichannel tunneling. 19.6. Reflection time in quantum tunneling. 19.7. Minimum tunneling time. 19.8. Traversal-time wave function -- 20. More about tunneling time. 20.1. Dwell and phase tunneling times. 20.2. Buttiker and Landauer time. 20.3. Larmor clock for measuring tunneling times. 20.4. Tunneling time and its determination using the internal energy of a simple molecule. 20.5. Intrinsic time. 20.6. Measurement of tunneling time by quantum clocks. 20.7. A critical study of the tunneling time determination by a quantum clock. 20.8. Tunneling time according to Low and Mende -- 21. Tunneling of a system with internal degrees of freedom. 21.1. Lifetime of coupled-channel resonances. 21.2. Two-coupled channel problem with spherically symmetric barriers. 21.3. Tunneling of a simple molecule. 21.4. Tunneling of a homonuclear molecule in a symmetric double-well potential. 21.5. Tunneling of a molecule in asymmetric double-wells. 21.6. Tunneling of a molecule through a potential barrier. 21.7. Tunneling of composite systems in nuclear reactions. 21.8. Antibound state of a molecule -- 22. Motion of a particle in a waveguide with variable cross section and in a space bounded by a dumbbell-shaped object. 22.1. An exactly solvable quantum waveguide. 22.2. Motion of a particle in a space bounded by a surface of revolution. 22.3. Testing the accuracy of the present method. 22.4. Calculation of the eigenvalues. 22.5. Quantum wires -- 23. Relativistic formulation of quantum tunneling. 23.1. One-dimensional tunneling of the electrons. 23.2. Relativistic effects in time-dependent tunneling. 23.3. Tunneling of spinless particles in one dimension. 23.4. Tunneling time in special relativity. 23.5. Quantum tunneling times for relativistic particles -- 24. Inverse problems of quantum tunneling. 24.1. A method for finding the potential from the reflection amplitude. 24.2. Determination of the shape of the potential barrier in one-dimensional tunneling. 24.3. Construction of a symmetric double-well potential from the known energy eigenvalues. 24.4. The inverse problem of tunneling for gamow states. 24.5. Prony's method for determination of complex energy eigenvalues -- 25. Some examples of quantum tunneling in atomic and molecular physics. 25.1. Torsional vibration of a molecule. 25.2 Electron emission from the surface of cold metals. 25.3. Ionization of atoms in very strong electric field. 25.4. A time-dependent formulation of ionization in an electric field. 25.5. Energy levels of the ammonia molecule and the ammonia maser. 25.6. Optical isomers. 25.7. Three-dimensional tunneling in the presence of a constant field of force -- 26. Some examples in condensed matter physics. 26.1. The band theory of solids and the Kronig-Penney model. 26.2. Tunneling in metal-insulator-metal structures. 26.3. Many-electron formulation of the current. 26.4. Excitation of closely spaced energy levels in heterostructures: The time-dependent formulation. 26.5. Electron tunneling through heterostructures. 26.6. The Josephson effect -- 27. Alpha decay. 27.1. The time-independent formulation of the [symbol] decay. 27.2. The time-dependent formulation of the [symbol] decay. 27.3. The WKB approximation. 27.4. Electromagnetic radiation by a charged particle while tunneling through a barrier. 27.5. Perturbation theory applied to the problem of Bremsstrahlung in [symbol]-decay. |

Responsibility: | Mohsen Razavy. |

### Abstract:

In this revised and expanded edition, in addition to a comprehensible introduction to the theoretical foundations of quantum tunneling based on different methods of formulating and solving tunneling problems, different semiclassical approximations for multidimensional systems are presented. Particular attention is given to the tunneling of composite systems, with examples taken from molecular tunneling and also from nuclear reactions. The interesting and puzzling features of tunneling times are given extensive coverage, and the possibility of measurement of these times with quantum clocks are critically examined. In addition, by considering the analogy between evanescent waves in waveguides and in quantum tunneling, the times related to electromagnetic wave propagation have been used to explain certain aspects of quantum tunneling times. These topics are treated in both non-relativistic as well as relativistic regimes. Finally, a large number of examples of tunneling in atomic, molecular, condensed matter and nuclear physics are presented and solved.

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