The Density Matrix Representation of Spin States

The effect of a radio frequency pulse is to mix the a and p states so that a superposition of states is obtained  

The diagonal terms will always be real numbers, and the a - p term ((cqdpav) will always be the complex conjugate of the p a term ((c2cpav). Now we can show the density matrix representations of the three single-spin product operators  

In all cases, the identity matrix and the factors NI2 and s have been omitted for simplicity. Note that the distinction between the x and y axes is made by using real and imaginary numbers for the off-diagonal terms. This is similar to the use of real and imaginary to represent the two parts of the FID signal (Mx and My) in the NMR spectrometer. A pulse can be represented by a matrix as well, so that the effect of any pulse can be calculated by simple matrix multiplication. For a general pulse of pulse angle , the rotation matrices are 

For example, starting from equilibrium (Iz), the effect of a 90° pulse on the y axis is 

The pulse is applied mathematically by multiplying the spin state matrix a by the rotation matrix R and then multiplying this result by the inverse matrix R l (the product R l R is the identity matrix 1). For rotation (pulse) operators, the inverse matrix is simply the rotation in the opposite direction ( = - ). Note that the final result is the same as the representation of the product operator x given above. 

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The density matrix representation of spin and orbital angular momentum is capable of expressing a static state of matter and its time-dependent response to an external perturbation. Our application necessitates that we follow the response of the orbital and spin momenta subject to full or partial excitations, and the density matrix provides a direct solution to the stochastic Liouville equation. But the density matrix representation in a rotating operator is algebraically ambiguious, and we must also clarify the algebraic description of selective excitation of multiquantum systems. 

Together, these 16 product operators describe the 16 matrix elements in the 4 x 4 density matrix representation of a two-spin system (Chapter 10). In the matrix, each element represents coherence between (or superposition of) two spin states. As there are four spin states for a two-spin system (aid s i/3s, P as, and PiPs), there are 16 possible pairs of states, which can be superimposed or share coherence. The product operators are closer to the visually and geometrically concrete vector model representations, so in most cases they are preferable to writing down the 16 elements of the density matrix, especially as only a few of the elements are nonzero in most of the examples we discuss. 

The purpose of this paper is to present the density matrix formalism of angular momentum with half- and integer spin quantum numbers using the spectrum dissolving theorem. The density matrix formalism was developed for the laboratory and rotating frame in order to obtain a complete analytical representation for the spin excitation and response scheme. The density matrix contained in the rotation operator will beeome elear, as oppossed to the approximate treatment, and thereby from the theoretieal process of off-resonance to that of just-resonance through the state of near-resonance will be visualized continuously. 

You will find that many of the sources do not use exactly the same matrix representations for some of the product operators and rotation matrices. The exact form of the density matrix depends on the numbering of the spin states and on certain conventions that are not consistent in the literature. In the above examples, the definitions are consistent with the product operator methods and with themselves. 

To facilitate the discussion, we couch DFT in the language of p, the first-order reduced density operator of the noninteracting reference system. Consider an N electron system in a spin-compensated state and in an external potential Wext(r) (extension to spin-polarized state is trivial). The real space representation of p is the density matrix... 

Most detailed studies of spin effects in the literature are based on the Stochastic Liouville equation (SLE), which treats the system as an ensemble and requires the use of the density matrix Pij(r, t) [7-9], The density matrix (i) contains all the information about the ensemble physical observables of the system (ii) describes the distribution of spin states for an ensemble of particles and (iii) is constructed from the vector representation of the spin function (c ) relative to some predefined basis, such that pij(r, t) = c Cj). 

Philip and Kuchel described a graphical representation of the spin states of quadrupolar nuclei in NMR experiments. The spherical harmonics represent the irreducible spherical tensor operator basis set, and therefore any density matrix can be illustrated with computer graphics by converting it to a sum of... 

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