Bound States - The University of Cincinnati, in Cincinnati, Ohio 束缚态-辛辛那提大学，辛辛那提，俄亥俄州.ppt

Bound States Physics 842, Winter 2007 B. Meadows What is a Bound State? Imagine a system of two bodies that interact. They can have relative movement. If this movement has sufficient energy, they will scatter and will eventually move far apart where their interaction will be negligible. If their interaction is repulsive, they will also scatter and move far apart to where their interaction is negligible. If the energy is small enough, and their interaction is attractive, they can become bound together in a “bound state”. In a bound state, the constituents still have relative movement, in general. If the interaction between constituents is repulsive, then they cannot form a bound state. Examples of bound states include: Atoms, molecules, positr-onium, prot-onium, quark-onium, mesons, baryons, … The Hydrogen Atom Proton-electron bound state described by the Schrodinger equation where Solutions are characterized by 3 quantum numbers (n,l,ml): where the Bohr radius and the Laguerre polynomials are The Hydrogen Atom Energy levels are: En = -? ?2mc2 /n2 = -13.6 /n2 (eV) (n=1,2,3,…) where ? = e2/~c = 1/137 is electro-magnetic coupling (“fine structure” constant) R = me4c/(4?~ 3) is the “Rydberg constant Radii of electron orbits are rn = an2 = 0.529 x 10-8 n2 Ionization potential (energy required to remove electron from n=1 to 1) is 13.6 eV. Quantum numbers: n = 1, 2, 3, 4, … l = 0, 1, … (n-1) m l = - l, -l+1,…, 0, … +l Perturbations Various effects change the energy levels Spin-orbit coupling Relativistic correction Hyperfine structure “Lamb shift” These are calculated by Griffiths (5.3 - 5.5) Each effect introduces a small correction to the Hamiltonian ?H that changes the energy Ej of state j by E.g. for the ground state Fine Structure – Spin-Orbit Coupling The electron spin s=?~ couples with its orbital angular momentum l ~ to make total spin j = (l § ?)~ The operators are S, L and J, respectively and J2 = L2 + S2 + 2 L ￠ S This spin-or