Most of the elementary quantum mechanical models-including particles in boxes, rigid rotor, harmonic oscillator, barrier penetration, hydrogen atom-are clearly and completely presented. All required math is clearly explained, including intermediate steps in derivations, and concise review of the math is included in the text at appropriate points. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.Download Fundamentals of Quantum Mechanics Book in PDF, Epub and Kindleįundamentals of Quantum Mechanics, Third Edition is a clear and detailed introduction to quantum mechanics and its applications in chemistry and physics. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. Finally, some aspects of the application of the invariants to quantum systems are discussed. The general solution for ρ(t) is evaluated for some special cases. The meaning of the invariants is discussed, and the general solution for ρ(t) is given in terms of linearly independent solutions of the equations of motion for the classical oscillator. The new coordinate is, of course, a cyclic variable. A generating function is given for a classical canonical transformation to a class of new canonical variables which are so chosen that the new momentum is any particular member of the class of invariants. As a consequence, the results are more general than the asymptotic treatment, and are even applicable with complex Ω(t) and quantum systems. The invariants are derived by applying an asymptotic theory due to Kruskal to the oscillator system in closed form. A class of exact invariants for oscillator systems whose Hamiltonians areis given in closed form in terms of a function ρ(t) which satisfies.Each particular solution of the equation for ρ determines an invariant.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |