Classical Mechanical Description of NMR

M0 is responsible for a small static nuclear paramagnetic susceptibility, but the nuclear susceptibility is virtually undetectable at room temperature in the presence of the diamagnetic susceptibility from paired electrons in the molecule, which is about four orders of magnitude larger. Although the static nuclear susceptibility has been detected for hydrogen at very low temperatures, this type of measurement is impractical. 

NMR provides a different and far superior means to measure M0 by tipping M away from the 2 axis into the xy plane, where it precesses at the Larmor frequency and can be selectively detected without interference from the static electron susceptibility. We can examine this process in classical terms. 

By multiplying both sides of this equation by y and using Eq. 2.2, we obtain 

Because the length of x is constant, Eq. 2.41 describes the rate at which the direction of p changes. If this change in direction is just a rotation with an angular momentum and direction given by a vector w1 the motion is described classically by 

The negative sign in Eq. 2.43 indicates that the vector to) that describes the rotation is directed opposite to B, a point that we take up further in Section 2.11. 

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Going beyond an atomistic description of the aqueous phase and the membrane, Paddison and coworkers [79-88] employed statistical mechanical models, incorporating solvent friction and spatially dependent dielectric properties, to the calculation of the proton diffusion coefficient in Nation and PEEKK membrane pores. They concluded from their studies that, in accordance with NMR based evidence [50], the mechanism of proton transport is more vehicular (classical ion transport) in the vicinity of the pore surface and more Grotthus-like in the center. 

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. 

The classical Liouville equation does have an equivalent in quantum mechanics, which is needed for a consistent description of quantum statistical mechanics the quantum Liouville equation. Equilibrium quantum statistical mechanics requires the introduction of the density operator on an appropriate Hilbert space, and the quantum liouvUle equation for the density operator is a logical and necessary extension of the Schrodinger equation. The quantum Liouville equation can even be written, formally at least, in a form that resembles its classical counterpart. It allows for some weak and almost internally consistent form of dissipative dynamics, known as the Redfield theory, which finds its main use in relating NMR relaxation times to spectral densities arising from solvent fluctuations, although in recent... 

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