Preface ......................................................... v
1 A Brief History of Quantum Tunneling ......................... 1
2 Some Basic Questions Concerning Quantum Tunneling ............ 9
2.1 Tunneling and the Uncertainty Principle ................. 9
2.2 Decay of a Quasistationary State ....................... 11
3 Semi-Classical Approximations ............................... 23
3.1 The WKB Approximation .................................. 23
3.2 Method of Miller and Good .............................. 31
3.3 Calculation of the Splitting of Levels in a Symmetric
Double-Well Potential Using WKB Approximation .......... 35
4 Generalization of the Bohr-Sommerfeld Quantization
Rule and its Application to Quantum Tunneling ............... 41
4.1 The Bohr-Sommerfeld Method for Tunneling in Symmetric
and Asymmetric Wells ................................... 45
4.2 Numerical Examples ..................................... 48
5 Gamow's Theory, Complex Eigenvalues, and the Wave Function
of a Decaying State ......................................... 53
5.1 Solution of the Schrodinger Equation with Radiating
Boundary Condition ..................................... 53
5.2 The Time Development of a Wave Packet Trapped Behind
a Barrier .............................................. 57
5.3 A More Accurate Determination of the Wave Function of
a Decaying State ....................................... 61
5.4 Some Instances Where WKB Approximation and the Gamow
Formula Do Not Work .................................... 66
6 Simple Solvable Problems .................................... 73
6.1 Confining Double-Well Potentials ....................... 73
6.2 Time-dependent Tunneling for a δ-Function Barrier ...... 79
6.3 Tunneling Through Barriers of Finite Extent ............ 82
6.4 Tunneling Through a Series of Identical Rectangular
Barriers ............................................... 90
6.5 Eckart's Potential ..................................... 96
6.6 Double-Well Morse Potential ............................ 99
7 Tunneling in Confining Symmetric and Asymmetric Double-
Wells ...................................................... 105
7.1 Tunneling When the Barrier is Nonlocal ................ 112
7.2 Tunneling in Separable Potentials ..................... 116
7.3 A Solvable Asymmetric Double-Well Potential ........... 119
7.4 Quasi-Solvable Examples of Symmetric and Asymmetric
Double-Wells .......................................... 121
7.5 Gel'fand-Levitan Method ............................... 124
7.6 Darboux's Method ...................................... 127
7.7 Optical Potential Barrier Separating Two Symmetric
or Asymmetric Wells ................................... 128
8 A Classical Description of Tunneling ....................... 139
9 Tunneling in Time-Dependent Barriers ....................... 149
9.1 Multi-Channel Schrodinger Equation for Periodic
Potentials ............................................ 150
9.2 Tunneling Through an Oscillating Potential Barrier .... 152
9.3 Separable Tunneling Problems with Time-Dependent
Barriers .............................................. 157
9.4 Penetration of a Particle Inside a Time-Dependent
Potential Barrier ..................................... 162
10 Decay Width and the Scattering Theory ...................... 167
10.1 Scattering Theory and the Time-Dependent Schrodinger
Equation .............................................. 168
10.2 An Approximate Method of Calculating the Decay
Widths ................................................ 173
10.3 Time-Dependent Perturbation Theory Applied to the
Calculation of Decay Widths of Unstable States ........ 176
10.4 Early Stages of Decay via Tunneling ................... 181
10.5 An Alternative Way of Calculating the Decay Width
Using the Second Order Perturbation Theory ............ 184
10.6 Tunneling Through Two Barriers ........................ 186
10.7 Escape from a Potential Well by Tunneling Through
both Sides ............................................ 191
10.8 Decay of the Initial State and the Jost Function ...... 196
11 The Method of Variable Reflection Amplitude Applied
to Solve Multichannel Tunneling Problems ................... 205
11.1 Mathematical Formulation .............................. 206
11.2 Matrix Equations and Semi-classical Approximation
for Many-Channel Problems ............................. 212
12 Path Integral and Its Semi-Classical Approximation in
Quantum Tunneling .......................................... 219
12.1 Application to the S-Wave Tunneling of a Particle
Through a Central Barrier ............................. 222
12.2 Method of Euclidean Path Integral ..................... 226
12.3 An Example of Application of the Path Integral
Method in Tunneling ................................... 231
12.4 Complex Time, Path Integrals and Quantum Tunneling .... 237
12.5 Path Integral and the Hamilton-Jacobi Coordinates ..... 241
12.6 Remarks About the Semi-Classical Propagator and
Tunneling Problem ..................................... 243
13 Heisenberg's Equations of Motion for Tunneling ............. 251
13.1 The Heisenberg Equations of Motion for Tunneling in
Symmetric and Asymmetric Double-Wells ................. 252
13.2 Tunneling in a Symmetric Double-Well .................. 258
13.3 Tunneling in an Asymmetric Double-Well ................ 259
13.4 Tunneling in a Potential Which Is the Sum of Inverse
Powers of the Radial Distance ......................... 261
13.5 Klein's Method for the Calculation of the
Eigenvalues of a Confining Double-Well Potential ...... 267
14 Wigner Distribution Function in Quantum Tunneling .......... 277
14.1 Wigner Distribution Function and Quantum Tunneling .... 281
14.2 Wigner Trajectory for Tunneling in Phase Space ........ 284
14.3 Wigner Distribution Function for an Asymmetric
Double-Well ........................................... 290
14.4 Wigner Trajectory for an Oscillating Wave Packet ...... 290
14.5 Margenau-Hill Distribution Function for a Double-
Well Potential ........................................ 292
15 Complex Scaling and Dilatation Transformation Applied
to the Calculation of the Decay Width ...................... 297
16 Multidimensional Quantum Tunneling ......................... 307
16.1 The Semi-classical Approach of Kapur and Peierls ...... 307
16.2 Wave Function for the Lowest Energy State ............. 311
16.3 Calculation of the Low-Lying Wave Functions by
Quadrature ............................................ 313
16.4 Method of Quasilinearization Applied to the Problem
of Multidimensional Tunneling ......................... 318
16.5 Solution of the General Two-Dimensional Problems ...... 323
16.6 The Most Probable Escape Path ......................... 327
17 Group and Signal Velocities ................................ 339
18 Time-Delay, Reflection Time Operator and Minimum
Tunneling Time ............................................. 351
18.1 Time-Delay in Tunneling ............................... 352
18.2 Time-Delay for Tunneling of a Wave Packet ............. 356
18.3 Landauer and Martin Criticism of theDefinition of
the Time-Delay in Quantum Tunneling ................... 365
18.4 Time-Delay in Multi-Channel Tunneling ................. 368
18.5 Reflection Time in Quantum Tunneling .................. 371
18.6 Minimum Tunneling Time ................................ 375
19 More about Tunneling Time .................................. 381
19.1 Dwell and Phase Tunneling Times ....................... 382
19.2 Buttiker and Landauer Time ............................ 385
19.3 Larmor Precession ..................................... 388
19.4 Tunneling Time and its Determination Using the
Internal Energy of a Simple Molecule .................. 392
19.5 Intrinsic Time ........................................ 394
19.6 A Critical Study of the Tunneling Time Determination
by a Quantum Clock .................................... 398
19.7 Tunneling Time According to Low and Mende ............. 402
20 Tunneling of a System with Internal Degrees of Freedom ..... 411
20.1 Lifetime of Coupled-Channel Resonances ................ 411
20.2 Two-Coupled Channel Problem with Spherically
Symmetric Barriers .................................... 413
20.3 A Numerical Example ................................... 415
20.4 Tunneling of a Simple Molecule ........................ 418
20.5 Tunneling of a Molecule in Asymmetric Double-Wells .... 424
20.6 Tunneling of a Molecule Through a Potential Barrier ... 429
20.7 Antibound State of a Molecule ......................... 434
21 Motion of a Particle in a Space Bounded by a Surface of
Revolution ................................................. 439
21.1 Testing the Accuracy of the Present Method ............ 444
21.2 Calculation of the Eigenvalues ........................ 445
22 Relativistic Formulation of Quantum Tunneling .............. 453
22.1 One-Dimensional Tunneling of the Electrons ............ 453
22.2 Tunneling of Spinless Particles in One Dimension ...... 458
22.3 Tunneling Time in Special Relativity .................. 461
23 The Inverse Problem of Quantum Tunneling ................... 471
23.1 A Method for Finding the Potential from the
Reflection Amplitude .................................. 472
23.2 Determination of the Shape of the Potential Barrier
in One-Dimensional Tunneling .......................... 473
23.3 Prony's Method of Determination of Complex Energy
Eigenvalues ........................................... 476
23.4 A Numerical Example ................................... 478
23.5 The Inverse Problem of Tunneling for Gamow States ..... 479
24 Some Examples of Quantum Tunneling in Atomic and
Molecular Physics .......................................... 485
24.1 Torsional Vibration of a Molecule ..................... 485
24.2 Electron Emission from the Surface of Cold Metals ..... 488
24.3 Ionization of Atoms in Very Strong Electric Field ..... 491
24.4 A Time-Dependent Formulation of Ionization in an
Electric Field ........................................ 493
24.5 Ammonia Maser ......................................... 497
24.6 Optical Isomers ....................................... 500
24.7 Three-Dimensional Tunneling in the Presence of
a Constant Field of Force ............................. 501
25 Examples from Condensed Matter Physics ..................... 511
25.1 The Band Theory of Solids and the Kronig-Penney
Model ................................................. 511
25.2 Tunneling in Metal-Insulator-Metal Structures ......... 515
25.3 Many Electron Formulation of the Current .............. 516
25.4 Electron Tunneling Through Hetero-structures .......... 525
26 Alpha Decay ................................................ 531
Index ......................................................... 541
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