The Quantum Mechanics Solver 2 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | XIV Contents 27 Bloch Oscillations .277 Unitary Transformation on a Quantum Band Structure in a Periodic The Phenomenon of Bloch Author Subject Index .289 Summary of Quantum Mechanics In the following pages we remind the basic definitions notations and results of quantum mechanics. 1 Principles Hilbert Space The first step in treating a quantum physical problem consists in identifying the appropriate Hilbert space to describe the system. A Hilbert space is a complex vector space with a Hermitian scalar product. The vectors of the space are called kets and are noted . The scalar product of the ket i and the ket l is noted 2 1 It is linear in 1 and antilinear in 2 and one has 1 2 2 1 Definition of the State of a System Pure Case The state of a physical system is completely defined at any time t by a vector of the Hilbert space normalized to 1 noted -0 t . Owing to the superposition principle if 1 and -02 are two possible states of a given physical system any linear combination W X C1 1 C2 2 where c1 and c2 are complex numbers is a possible state of the system. These coefficients must be chosen such that 1. 2 Summary of Quantum Mechanics Measurement To a given physical quantity A one associates a self-adjoint or Hermitian operator A acting in the Hilbert space. In a measurement of the quantity A the only possible results are the eigenvalues aa of A. Consider a system in a state -0 . The probability P aa to find the result aa in a measurement of A is P aa P W 2 where Pa is the projector on the eigensubspace Ea associated to the eigenvalue aa. After a measurement of A which has given the result aa the state of the system is proportional to Pa wave packet projection or reduction . A single measurement gives information on the state of the system after the measurement has been performed. The information acquired on the state before the measurement is very poor . if .
