tailieunhanh - Basic Theoretical Physics: A Concise Overview P34

Basic Theoretical Physics: A Concise Overview P34. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 44 Statistical Physics Introduction Boltzmann-Gibbs Probabilities Consider a quantum mechanical system confined in a large volume for which all energy values are discrete. Then Hi Ej I where H is the Hamilton operator and I the complete set of orthogonal normalized eigenfunctions of H. The observables of the system are represented by Hermitian1 operators A . with real eigenvalues. For example the spatial representation of the operator p is given by the differential operator p - V i and the space operator r by the corresponding multiplication operator. h h 2n is the reduced Planck s constant and i the square root of minus one the imaginary unit . The eigenvalues of A are real as well as the expectation values 1 I-4 1 which are the averages of the results of an extremely comprehensive series of measurements of A in the state ij. One can calculate these expectation values . primary experimental quantities theoretically via the scalar product given above for example in the one-particle spatial representation without spin as follows I 4 ij y d3rl r t A p r r t where 1 M y d3rl r t lj r t 1 The thermal expectation value at a temperature T is then A t pi 1 IAl 1 more precisely by self-adjoint operators which are a Hermitian and b possess a complete system of eigenvectors. 344 44 Statistical Physics with the Boltzmann-Gibbs probabilities _ Ej e kBT Pj Ei . 2 e kBT i Proof of the above expression will be deferred since it relies on a so-called microcanonical ensemble and the entropy of an ideal gas in this ensemble will first be calculated. In contrast to quantum mechanics where probability amplitudes are added . ci i C2 2 the thermal average is incoherent. This follows explicitly from . This equation has the following interpretation the system is with probability pj in the quantum mechanical pure state j2 and the related quantum mechanical expectation values for j 1 2 . which are bilinear expressions in fy are then added like intensities as in .