دانلود کتاب فیزیک نظری فشرده III – مکانیک کوانتومی

This book provides a textbook on quantum mechanics and is in particular suited for bachelor students in their second or third year of studies in theoretical physics. In Part I a short summary of classical mechanics and its limits is given by pointing out those physical observations, that are in conlict with (or opposed to) interpretations within classical mechanics. Early attempts to extend the classical physics by additional boundary conditions are discussed as well. In Part II the Elementary Quantum Mechanics for a single particle is introduced and the statistical interpretation of its wave function, which results from the Schrödinger equation in the presence of external forces. It is shown how to translate classical observables in phase-space representation to quantum mechanical operators acting in an abstract Hilbert space of wave functions. Observable quantities like momentum, angular momentum or energy then are deined by expectation values of their corresponding operators. Contrary to classical mechanics the sequence of measurements on different observables in general cannot be exchanged; this is relected in commutators between operators that have the same algebra as the Poisson brackets in classical mechanics. A prominent example is the uncertainty relation between position and momentum, which states that the product of the uncertainty in position and the uncertainty in momentum has a lower limit in quantum mechanics and is increasing in time for free wave packets. Another example is the angular momentum, whose components do not commute and relect the fact that the order of rotations cannot be exchanged. In line with the observations in the Stern-Gerlach experiments a spin is attributed to electrons and described by an internal degree of freedom with the same algebra as the angular momentum. This inally results in the Pauli equation that describes the dynamics of charged spin 1/2 particles e.g. in an external electromagnetic ield. The general properties of quantum mechanics then are illustrated by a couple of examples including the ininite (and inite) square well, the quantum mechanical tunnelling through a inite potential well as well as problems with crystal (periodic) symmetries leading to inite energy bands in solid state materials. The problem of the harmonic oscillator is solved algebraically and its spectrum is derived as well as its eigenfunctions in one and three dimensions. In case of radial symmetry the radial Schrödinger equation is derived and solved explicitly for the case of the hydrogen atom. Apart from bound states of the single-particle problem, furthermore, continuum states are investigated in the framework of scattering theory and an alternative formulation of quantum mechanics is found in terms of the Lippmann- Schwinger equation, which is more appropriate for scattering problems due to the explicit implementation of proper boundary conditions. In this context the scattering amplitude and the differential cross section are introduced, that describe the scattering probability in quantum theory. The Born series is set up for the scattering amplitude and investigated in leading order approximation. Finally, the scattering amplitude is derived in angular momentum representation and scattering phase shifts are introduced, that deine the S-matrix for elastic scattering (and ixed angular momentum). The Mathematical Foundations of Quantum Mechanics are addressed in Part III of this book that aims at a rigid formulation of many-particle problems in quantum physics. To this aim in particular self-adjoint, unitary and projection operators are discussed in detail. In this context the possible many-body states are characterized with respect to their particle-exchange symmetry (in case of identical particles) which leads to the classiication of fermions and bosons, which differ by a minus- sign in their wave function for the exchange of two particles. The Pauli principle, furthermore, states that fermions can only occupy a state (with given quantum numbers) once, whereas there is no limitation for bosons. This distinction is not possible in classical mechanics, since the particles can be distinguished by their trajectories in phase space, but has severe consequences for many-body systems close to their ground state.