Density Operator Formalism

In the density operator formalism the actual state of an ensemble of spins is characterized by a density operator σ. The properties of this density operator facilitate the description of spin dynamics in pulse EPR and NMR considerably.

The density operator σ is a hermitian operator and can be represented by a matrix of dimension dHS×dHS. In the eigenbasis of Hsp the diagonal elements of σ are the populations of the corresponding eigenstate of Hsp whereas off-diagonal elements are coherent superpositions of eigenstates which can be related to the transverse magnetization Mx ± iMy (single-quantum coherence). The time evolution of the system is given by the Liouville-von-Neumann equation

$$\frac{d\sigma}{dt} = -i \left[\mathcal{H}_\mathrm{sp} , \sigma \left({t}\right) \right]\qquad (A. 2) $$

For a time-independent Hamiltonian (e.g. undisturbed spin system in the absence of MW pulses), integration of the Liouville-von-Neumann equation yields the solution

$$\sigma \left( t \right) = e^{-it\mathcal{H}_\mathrm{sp}}\sigma \left( 0 \right) e^{it\mathcal{H}_\mathrm{sp}}\qquad (A. 3) $$

With this equation the evolution of the spin system under different Hamiltonians and for different time intervals t can be calculated in subsequent steps which is very convenient for the theoretical description of complex pulse experiments.

The expectation value of an arbitrary observable operator P is found by evaluating the trace of the product of the observable operator P with the density operator σ

$$ \langle{P}\rangle = \mathrm{Tr} \left\{ P \sigma \left( t \right) \right\} \qquad (A. 4) $$

The description of pulse EPR experiments can be highly formalized by expressing the density operator and the Hamiltonian by products of the cartesian spin operators Sx, Sy, Sz and the unit operator. In this formalism equilibrium magnetization is written as Sz, while the operators Sx or Sy contribute to the observable magnetization in the xy-plane.

JavaScript has been disabled in your browser