Quantum Theory of Tunneling

by Razavy, Mohsen

• ISBN13: 9789812380180
• ISBN10: 9812380183
• Format: Hardcover
• Copyright: 2003-04-01
• Publisher: World Scientific Pub Co Inc

Summary

This book provides a comprehensive introduction to the theoretical foundations of quantum tunneling, stressing the basic physics underlying the applications. The topics addressed include exponential and nonexponential decay processes and the application of scattering theory to tunneling problems. In addition to the Schrvdinger equation approach, the path integral, Heisenberg's equations and the phase space method are all used to study the motion of a particle under the barrier. Extensions to the multidimensional cases and tunneling of particles with internal degrees of freedom are also considered. Furthermore, recent advances concerning time delay and tunneling times and some of the problems associated with their measurement are also discussed. Finally, some examples of tunneling in atomic, molecular, nuclear and condensed matter physics are presented.

Table of Contents

Preface	p. v
A Brief History of Quantum Tunneling	p. 1
Some Basic Questions Concerning Quantum Tunneling	p. 9
Tunneling and the Uncertainty Principle	p. 9
Decay of a Quasistationary State	p. 11
Semi-Classical Approximations	p. 23
The WKB Approximation	p. 23
Method of Miller and Good	p. 31
Calculation of the Splitting of Levels in a Symmetric Double-Well Potential Using WKB Approximation	p. 35
Generalization of the Bohr-Sommerfeld Quantization Rule and its Application to Quantum Tunneling	p. 41
The Bohr-Sommerfeld Method for Tunneling in Symmetric and Asymmetric Wells	p. 45
Numerical Examples	p. 48
Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State	p. 53
Solution of the Schrodinger Equation with Radiating Boundary Condition	p. 53
The Time Development of a Wave PacketTrapped Behind a Barrier	p. 57
A More Accurate Determination of the Wave Function of a Decaying State	p. 61
Some Instances Where WKB Approximation and the Gamow Formula Do Not Work	p. 66
Simple Solvable Problems	p. 73
Confining Double-Well Potentials	p. 73
Time-dependent Tunneling for a [delta]-Function Barrier	p. 79
Tunneling Through Barriers of Finite Extent	p. 82
Tunneling Through a Series of Identical Rectangular Barriers	p. 90
Eckart's Potential	p. 96
Double-Well Morse Potential	p. 99
Tunneling in Confining Symmetric and Asymmetric Double-Wells	p. 105
Tunneling When the Barrier is Nonlocal	p. 112
Tunneling in Separable Potentials	p. 116
A Solvable Asymmetric Double-Well Potential	p. 119
Quasi-Solvable Examples of Symmetric and Asymmetric Double-Wells	p. 121
Gel'fand-Levitan Method	p. 124
Darboux's Method	p. 127
Optical Potential Barrier Separating Two Symmetric or Asymmetric Wells	p. 128
A Classical Description of Tunneling	p. 139
Tunneling in Time-Dependent Barriers	p. 149
Multi-Channel Schrodinger Equation for Periodic Potentials	p. 150
Tunneling Through an Oscillating Potential Barrier	p. 152
Separable Tunneling Problems with Time-Dependent Barriers	p. 157
Penetration of a Particle Inside a Time-Dependent Potential Barrier	p. 162
Decay Width and the Scattering Theory	p. 167
Scattering Theory and the Time-Dependent Schrodinger Equation	p. 168
An Approximate Method of Calculating the Decay Widths	p. 173
Time-Dependent Perturbation Theory Applied to the Calculation of Decay Widths of Unstable States	p. 176
Early Stages of Decay via Tunneling	p. 181
An Alternative Way of Calculating the Decay Width Using the Second Order Perturbation Theory	p. 184
Tunneling Through Two Barriers	p. 186
Escape from a Potential Well by Tunneling Through both Sides	p. 191
Decay of the Initial State and the Jost Function	p. 196
The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems	p. 205
Mathematical Formulation	p. 206
Matrix Equations and Semi-classical Approximation for Many-Channel Problems	p. 212
Path Integral and Its Semi-Classical Approximation in Quantum Tunneling	p. 219
Application to the S-Wave Tunneling of a Particle Through a Central Barrier	p. 222
Method of Euclidean Path Integral	p. 226
An Example of Application of the Path Integral Method in Tunneling	p. 231
Complex Time, Path Integrals and Quantum Tunneling	p. 237
Path Integral and the Hamilton-Jacobi Coordinates	p. 241
Remarks About the Semi-Classical Propagator and Tunneling Problem	p. 243
Heisenberg's Equations of Motion for Tunneling	p. 251
The Heisenberg Equations of Motion for Tunneling in Symmetric and Asymmetric Double-Wells	p. 252
Tunneling in a Symmetric Double-Well	p. 258
Tunneling in an Asymmetric Double-Well	p. 259
Tunneling in a Potential Which Is the Sum of Inverse Powers of the Radial Distance	p. 261
Klein's Method for the Calculation of the Eigenvalues of a Confining Double-Well Potential	p. 267
Wigner Distribution Function in Quantum Tunneling	p. 277
Wigner Distribution Function and Quantum Tunneling	p. 281
Wigner Trajectory for Tunneling in Phase Space	p. 284
Wigner Distribution Function for an Asymmetric Double-Well	p. 290
Wigner Trajectory for an Oscillating Wave Packet	p. 290
Margenau-Hill Distribution Function for a Double-Well Potential	p. 292
Complex Scaling and Dilatation Transformation Applied to the Calculation of the Decay Width	p. 297
Multidimensional Quantum Tunneling	p. 307
The Semi-classical Approach of Kapur and Peierls	p. 307
Wave Function for the Lowest Energy State	p. 311
Calculation of the Low-Lying Wave Functions by Quadrature	p. 313
Method of Quasilinearization Applied to the Problem of Multidimensional Tunneling	p. 318
Solution of the General Two-Dimensional Problems	p. 323
The Most Probable Escape Path	p. 327
Group and Signal Velocities	p. 339
Time-Delay, Reflection Time Operator and Minimum Tunneling Time	p. 351
Time-Delay in Tunneling	p. 352
Time-Delay for Tunneling of a Wave Packet	p. 356
Landauer and Martin Criticism of the Definition of the Time-Delay in Quantum Tunneling	p. 365
Time-Delay in Multi-Channel Tunneling	p. 368
Reflection Time in Quantum Tunneling	p. 371
Minimum Tunneling Time	p. 375
More about Tunneling Time	p. 381
Dwell and Phase Tunneling Times	p. 382
Buttiker and Landauer Time	p. 385
Larmor Precession	p. 388
Tunneling Time and its Determination Using the Internal Energy of a Simple Molecule	p. 392
Intrinsic Time	p. 394
A Critical Study of the Tunneling Time Determination by a Quantum Clock	p. 398
Tunneling Time According to Low and Mende	p. 402
Tunneling of a System with Internal Degrees of Freedom	p. 411
Lifetime of Coupled-Channel Resonances	p. 411
Two-Coupled Channel Problem with Spherically Symmetric Barriers	p. 413
A Numerical Example	p. 415
Tunneling of a Simple Molecule	p. 418
Tunneling of a Molecule in Asymmetric Double-Wells	p. 424
Tunneling of a Molecule Through a Potential Barrier	p. 429
Antibound State of a Molecule	p. 434
Motion of a Particle in a Space Bounded by a Surface of Revolution	p. 439
Testing the Accuracy of the Present Method	p. 444
Calculation of the Eigenvalues	p. 445
Relativistic Formulation of Quantum Tunneling	p. 453
One-Dimensional Tunneling of the Electrons	p. 453
Tunneling of Spinless Particles in One Dimension	p. 458
Tunneling Time in Special Relativity	p. 461
The Inverse Problem of Quantum Tunneling	p. 471
A Method for Finding the Potential from the Reflection Amplitude	p. 472
Determination of the Shape of the Potential Barrier in One-Dimensional Tunneling	p. 473
Prony's Method of Determination of Complex Energy Eigenvalues	p. 476
A Numerical Example	p. 478
The Inverse Problem of Tunneling for Gamow States	p. 479
Some Examples of Quantum Tunneling in Atomic and Molecular Physics	p. 485
Torsional Vibration of a Molecule	p. 485
Electron Emission from the Surface of Cold Metals	p. 488
Ionization of Atoms in Very Strong Electric Field	p. 491
A Time-Dependent Formulation of Ionization in an Electric Field	p. 493
Ammonia Maser	p. 497
Optical Isomers	p. 500
Three-Dimensional Tunneling in the Presence of a Constant Field of Force	p. 501
Examples from Condensed Matter Physics	p. 511
The Band Theory of Solids and the Kronig-Penney Model	p. 511
Tunneling in Metal-Insulator-Metal Structures	p. 515
Many Electron Formulation of the Current	p. 516
Electron Tunneling Through Hetero-structures	p. 525
Alpha Decay	p. 531
Index	p. 541
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