Hypervirial Theorems

ISBN-13: 9783540171706 / Angielski / Miękka / 1987 / 373 str.

Francisco M. Fernandez; Eduardo Alberto Castro

Spis treści

A.- I. Hypervirial Theorems and Exact Solutions of the Schrödinger Equation.- 1. Equations of motion.- 2. Diagonal hypervirial theorems.- 3. Off-diagonal hypervirial theorems.- 4. Quantum-mechanical sum rules.- 5. Recurrence relations among matrix elements of functions of the coordinate.- 6. Hypervirial theorems for unbound states.- 7. Quantum-mechanical virial theorem.- 8. Derivatives of the energy with respect to a parameter in the Hamiltonian operator.- References Chapter I.- II. Hypervirial Theorems and Perturbation Theory.- 9. Ray1eigh-Schrödinger perturbation theory.- 10. Perturbation theory and hypervirial theorems.- References Chapter II.- III. Hypervirial Theorems and the Variational Theorem.- 11. Variational theorem.- 12. Unitary operator formalism.- 13. Point transformation formalism.- 14. Hellmann-Feynman theorem and variational functions.- 15. Simultaneous hypervirial relationships.- 16. Hypervirial relations and symmetry conditions.- 17. Tensorial generalization of the quantum virial theorem.- 18. Coordinate shifting (Translation).- 19. Some worked examples illustrating the application of the several transformations.- 20. Linear transformation and correlation of variables.- Numerical results.- References Chapter III.- IV. Non Diagonal Hypervirial Theorems and Approximate Functions.- 21. Fundamental theorems.- 22. The non diagona1-diagona1 hypervirlal iterative method.- 23. Sum rules and approximate functions.- 24. Non diagonal hypervirial theorems and approximate functions.- 25. Hypervirial theorems and orthogonality conditions.- 26. Restricted variational method.- References Chapter IV.- V. Hypervirial Functions and Self-Consistent Field Functions.- 27. Self-consistent functions. Hartree Method.- 28. Self-consistent function for identical particles. Hartree-Fock Method.- Numerical results.- References Chapter V.- VI. Perturbation Theory Without Wave Function.- 29. 1D Models.- 30. Central potential systems.- 31. 1D systems with periodic potentials.- Numerical results.- References Chapter VI.- B.- VII. Importance of the Different Boundary Conditions.- 32. Boundary conditions.- 33. Hypervirial theorems for finite boundary conditions.- References Chapter VII.- VIII. Hypervirial Theorems for 1D Finite Systems. General Boundary Conditions.- 34. Reformulation of some theorems.- 35. Hypervirial theorems for 1D systems under general BC.- References Chapter VIII.- IX. Hypervirial Theorems for 1D Finite Systems. Dirichlet Boundary Conditions.- 36. General equations and sum rules.- 37. Simple models with Dirichlet Boundary conditions.- 38. Symmetrical oscillators — DBC.- 39. Harmonic oscillator and DBC.- Numerical results.- References Chapter IX.- X. Hypervirial Theorems for Finite 1D Systems. Von Neumann Boundary Conditions.- 40. General equations.- 41. Bounded oscillators.- 42. Semi-infinite systems and periodic potentials.- Numerical results.- References Chapter X.- XI. Hypervirial Theorems for Finite Multidimensional Systems.- 43. General equations.- 44. Dirichlet boundary conditions.- 45. Von Neumann boundary conditions.- Numerical results.- References Chapter XI.- Special Topics.- 46. Hypervirial theorems and statistical quantum mechanics.- 47. Hypervirial theorems and semiclassica1 approximation.- Numerical results.- References.- Appendix I. Evolution operators.- Appendix II. Hamiltonian of an isolated N-particles system.- Appendix III. Project ion operators.- Appendix IV. Perturbation theory.- Appendix V. Differentiation of matrices and determinants.- Apendix VI. Dynamics of systems with time independent Hamiltonians.- Appendix VII. Elements of probability theory for continuous random variables.- Appendix VIII. Electrons in crystal lattices.- Appendix IX. Numerical integration of the Schrödinger equation.- Appendix X. Expansion in cthz series and polynomial power coefficients.- Bibliography and References for Appendices.- Program I.- Program II.- Program III.- Program IV.- Program V.- Program VI.- Program VII.- Program VIII.- Program IX.- Program X.- Program XI.- Program XII.- Program XIII.- Program XIV.- Program XV.- Program XVI.- Program XVII.