tailieunhanh - The Quantum Mechanics Solver 9

The Quantum Mechanics Solver 9 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | Decay of Positronium 75 . a Taking into account the result of question and setting w -yB write the action of HZ on the basis states o1 ct2 . b Write in terms of A and hw the matrix representation of H Hss Ha Hz in the basis S m of the total spin of the two particles. c Give the numerical value of hw in eV for a field B 1 T. Is it easy experimentally to be in a strong field regime . hw A . Calculate the energy eigenvalues in the presence of the field B express the corresponding eigenstates in the basis S m of the total spin. The largest eigenvalue will be written E and the corresponding eigenstate -0 . For convenience one can introduce the quantity x 8hw 7A and the angle 3 defined by sin 20 x 1 x2 cos 20 1 1 x2. . Draw qualitatively the variations of the energy levels in terms of B. Are there any remaining degeneracies Decay of Positronium We recall that when a system A is unstable and decays A B the probability for this system to decay during the interval t t dt if it is prepared at t 0 is dp Ae-Xtdt where the decay rate A is related to the lifetime t of the system by t 1 A. If the decay can proceed via different channels . A B and A C with respective decay rates Ai and A2 the total decay rate is the sum of the partial rates and the lifetime of A is t 1 A1 A2 . In all what follows we place ourselves in the rest frame of the positronium. . In a two-photon decay or annihilation of positronium what are the energies of the two outgoing photons and what are their relative directions . One can show that the annihilation rate of positronium into photons in an orbital state n l m is proportional to the probability for the electron and positron to be at the same point . to nim 0 2. In what orbital states is the annihilation possible . In quantum field theory one can show that owing to charge conjugation invariance a a singlet state S 0 can only decay into an even number of photons 2 4 b a triplet state S 1 can only .