Hamiltonian equations relaxation mechanisms

The dynamics of nuclear spins can be treated by a time-dependent Schrodinger equation, where the Hamiltonian contains terms for each constituting relaxation mechanism ... 

Now H t) in Equation [9] determines what is called the spin relaxation mechanism. As an example, the dipole-dipole Hamiltonian or quadrupolar Hamiltonian with an axially symmetric ( = 0) electric field gradient tensor is given by... 

It seems that the conventional approach to the quantum mechanical master equation relies on the equilibrium correlation function. Thus the CTRW method used by the authors of Ref. 105, yielding time-convoluted forms of GME [96], can be made compatible with the GME derived from the adoption of the projection approach of Section III only when p > 2. The derivation of this form of GME, within the context of measurement processes, was discussed in Ref. 155. The authors of Ref. 155 studied the relaxation process of the measurement pointer itself, described by the 1/2-spin operator Ez. The pointer interacts with another 1/2-spin operator, called av, through the interaction Hamiltonian... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

As soon as large ensembles of particles with statistical populations of the eigenstates and incoherent exchange and relaxation processes between these states are investigated, quantum statistical tools are necessary to describe the system. In this situation the quantum mechanical density operator p has to be employed. For the coherent evolution of the density operator under the influence of a Hamiltonian H, the following differential equation is found [80]... 

Hamiltonian matrix elements, write an expression for the time dependence of each element of p (e.g., dp ldt) in the absence of stochastic relaxations, (c) What is the relationship between / ab(0 and / ba(0 (d) Suppose that interconversions of the two basis states are driven only by the quantum mechanical coupling element but that stochastic fluctuations of the energies cause pure dephasing with a time constant 7. What are the longitudinal (Ji) and transverse (T2) relaxation times in this situation (e) Write out the Stochastic Liouville expression for the time-dependence of each element of p. (/) How would Ti and 7 2 be modified if the system also changes stochastically from state atob with rate constant and from btoa with rate constant ba ( ) I what limit does the stochastic Liouville equation reduce to the golden-rule expression ... 

---