Quantum mechanical density operator coherences

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]... 

In our introduction to the physics of NMR in Chapter 2, we noted that there are several levels of theory that can be used to explain the phenomena. Thus far we have relied on (1) a quantum mechanical treatment that is restricted to transitions between stationary states, hence cannot deal with the coherent time evolution of a spin system, and (2) a picture of moving magnetization vectors that is rooted in quantum mechanics but cannot deal with many of the subder aspects of quantum behavior. Now we take up the more powerful formalisms of the density matrix and product operators (as described very briefly in Section 2.2), which can readily account for coherent time-dependent aspects of NMR without sacrificing the quantum features. 

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... 

The detection of NMR signals is based on the perturbation of spin systems that obey the laws of quantum mechanics. The effect of a single hard pulse or a selective pulse on an individual spin or the basic understanding of relaxation can be illustrated using a classical approach based on the Bloch equations. However as soon as scalar coupling and coherence transfer processes become part of the pulse sequence this simple approach is invalid and fails. Consequently most pulse experiments and techniques cannot be described satisfactorily using a classical or even semi-classical description and it is necessary to use the density matrix approach to describe the quantum physics of nuclear spins. The density matrix is the basis of the more practicable product operator formalism. 

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