## Exercises in Quantum Mechanics: A Collection of Illustrative Problems and Their SolutionsThis monograph is written within the framework of the quantum mechanical paradigm. It is modest in scope in that it is restricted to some observations and solved illustrative problems not readily available in any of the many standard (and several excellent) texts or books with solved problems that have been written on this subject. Additionally a few more or less standard problems are included for continuity and purposes of comparison. The hope is that the points made and problems solved will give the student some additional insights and a better grasp of this fascinating but mathematically somewhat involved branch of physics. The hundred and fourteen problems discussed have intentionally been chosen to involve a minimum of technical complexity while still illustrating the consequences of the quantum-mechanical formalism. Concerning notation, useful expressions are displayed in rectangular boxes while calculational details which one may wish to skip are included in square brackets. Beirut HARRY A. MAVROMATIS June, 1985 IX Preface to Second Edition More than five years have passed since I prepared the first edition of this mono graph. The present revised edition is more attractive in layout than its predecessor, and most, if not all of the errors in the original edition (many of which were kindly pointed out by reviewers, colleagues, and students) have now been corrected. Additionally the material in the original fourteen chapters has been extended with significant additions to Chapters 8, 13, and 14. |

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### Contents

WilsonSommerfeld Quantization Condition | 1 |

The Delta Function Completeness and Closure | 16 |

Momentum Space | 34 |

Wavepackets and the Uncertainty Principle | 47 |

Uncertainty Principle and GroundState Energies of QuantumMechanical Systems | 68 |

Free Particles Incident on Potentials Time Delay Phase Shifts and The Born Approximation | 74 |

Heisenberg Representation | 102 |

Two Three and N Versus OneDimensional Problems | 111 |

Perturbation Theory | 157 |

Degeneracy | 186 |

The Inverse Problem | 214 |

The DalgarnoLewis Technique | 227 |

Angular Momentum and Coupled States | 262 |

Tensor Operators and Evaluation of Matrix Elements | 291 |

Applications of Quantum Mechanics | 312 |

Index | 326 |

Kramers Type Expressions The Virial Theorem and Generalizations | 124 |

Upper Bounds and Parity Considerations | 133 |

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Exercises in Quantum Mechanics: A Collection of Illustrative Problems and ... H.A. Mavromatis No preview available - 2012 |

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Aground exact angular momentum coupling angular momentum operator antisymmetric Born approximation Chapter choice complete set Consider a particle Consider the Hamiltonian coordinates corresponding Coulomb potential coupling coefficients cross section Dalgarno-Lewis degeneracy degenerate delta function dimensional eigenfunctions eigenvalue energy levels evaluate exact energy exact ground-state expansion expression Find the energy ground-state energy Hamiltonian Hence implies infinite integral involves J-oo matrix element Mavromatis momentum space normalized notes obtains odd-parity one-dimensional optimal parity considerations perturbation theory phase shifts Potential in Example problem Quantum Mechanics quantum number representation result Eq results of Examples satisfy scattering amplitude Schrödinger equation second-order simple harmonic oscillator solutions solved spherical harmonics substituting in Eq symmetric Table Tables of Integrals three-dimensional trial wave function two-dimensional unperturbed values wavepacket Wigner Wigner-Eckart theorem yields zero Фо

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Page iii - Department of Physics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Kluwer Academic Publishers Dordrecht / Boston / London by • "CIP-data available from the Publisher on request.