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The Quantum Mechanics Solver 25

The Quantum Mechanics Solver 25 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 | 246 24 Probing Matter with Positive Muons m.c 105.66 MeV a1 h 67.5 MHz T mec2 0.511 MeV p2 h -1.40 x 104 MHz T . 24.1 Muonium in Vacuum Muonium is formed by slowing down a beam of a prepared in a given spin state in a thin metal foil. A sufficiently slow a can capture an electron and form a hydrogen-like atom in an excited state. This atom falls into its ground state very quickly in 10 _9 s the muon s spin state remaining the same during this process. Once it is formed the muonium which is electrically neutral can diffuse outside the metal. We assume that at t 0 the state of the muonium atom is the following The muon spin is in the eigenstate z of j1z. The electron spin is in an arbitrary state a with a 2 0 2 1. The wave fuction r of the system is the 1s wave function of the hydrogen-like system ioo r . Just as for the hyperfine structure of hydrogen we work in the 4 dimensional Hilbert space corresponding to the spin variables of the electron and the muon. In this Hilbert space the spin-spin interaction Hamiltonian is H Eo - 4n ioo 0 2 Mi M2 Eo A i 2 where the indices 1 and 2 refer respectively to the muon and to the electron and where Eo mrc2a2 2 with mr being the reduced mass of the e a system. 24.1.1. Write the matrix representation of the Hamiltonian H in the basis 1z 2z iz . 24.1.2. Knowing the value of A in the hydrogen atom A h 1420 MHz calculate A in muonium. We recall that p1 qh 2m for the muon p2 qh 2me for the electron pp 2.79 qh 2mp for the proton where q is the unit charge and mp 1816.1 me. 24.1.3. Write the general form of an eigenstate of a1z with eigenvalue 1 i in the basis 1z 2z ii in the eigenbasis of H. 24.1.4. We assume that at t 0 the system is in a state 0 of the type defined above. Calculate t at a later time. 24.1.5. a Show that the operators n 1 j1z 2 are the projectors on the eigenstates of 1z corresponding to the eigenvalues 1. b Calculate for the state t the probability p t that the muon spin is in the state at time t. Write the result