Density matrix at equilibrium

As we shall see, it is very helpful to be able to compute the behavior of the spin system from the sort of matrix multiplications that we have already carried out. On the other hand, it is often possible to simplify the algebraic expressions by using the corresponding spin operators. In fact, this is the concept of the product operator formalism that we discuss later. Note that from Eqs. 11.35 and 11.36, the (redefined) density matrix at equilibrium can be written in operator form as... 

With the approximations used in Eq. 11.33, we can represent the relative populations of the four energy levels in terms of e, = (E3 — Ex)/kT and es = (E2 — E /kT. We already know that the density matrix at equilibrium p(0) is diagonal, with elements that describe the excess populations ... 

We saw in previous sections that it is often possible to express the density matrix for a particular situation in a more concise operator form. For example, in Eq. 11.37 we saw that the density matrix at equilibrium is given by (e/2)Iz, and in Eq. 11.57 we showed that application of a nonselective 90 x pulse to a two-spin system generated a density matrix that could be expressed as (e/2)Iy + (es/2)Sy. To make the expressions even simpler, we can drop the normalizing factors and for heteronuclear systems incorporate the relative magnitudes given by the e s into the spin operators themselves. For this example, we then obtain simply Iy + Sr... 

To work with Eq. (257) we need a p t ) in which we have confidence. On the understanding that we are dealing only with processes near equilibrium, we should choose a p(p ) similar in form to the density matrix at equilibrium, p. It is well known that p has the canonical form ... 

Trexp(—/ if) is the partition function. The response of the system to the external force in a property B is given by the average value of the operator B using the density matrix at time t (assuming, without loss of generality, that B has an average value of zero in equilibrium),... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

We start at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the weighted sum of the operators, which we will call 4- We assume that a simple, non-selective pulse has been used at the start of the experiment. This rotates the equilibrium 2 magnetization onto the x axis. After the pulse the density matrix is therefore given by 4, and it will evolve as in equation (7) or (8). If we substitute (8) into (10), we get the NMR signal as a function of time t, as given by (11). The detector sees each spin (but not each coherence ) equally well. 

Solve the Liouville equation for the density matrix p(0 at time t, given that the system is initially in thermodynamic equilibrium. 

It is assumed that the system is in a state of thermodynamic equilibrium at temperature T prior to the application of the forces Fj. j5( —oo) must consequently be the canonical density matrix, p0,... 

C. Spin density matrix of a spin system at thermal equilibrium with a lattice. 231... 

A spin system placed in a constant homogeneous magnetic field B0 finally attains thermal equilibrium with the lattice. This relaxation is induced by fluctuations of local magnetic fields which result from molecular motions. At equilibrium, the spin system is described by the equilibrium density matrix p0 ... 

The spin density matrix Pj(t) which describes the properties of any spin system of a molecule A, is defined as follows. We assume that the density matrices Pj(0), j = 1, 2,..., S, which describe the individual components of the dynamic equilibrium at any arbitrary time zero, are known explicitly, and that at any time t such that t > t > 0 the pj(t ) matrices are already defined. Our reasoning is applied to a pulse-type NMR experiment, and we therefore construct the equation of motion in a static magnetic field. The p,(t) matrix is the weighted average over the states involved, according to equation (5). The state of a molecule A, formed at the moment t and persisting as such until t, is given by the solution of equation (35) with the super-Hamiltonian H° ... 

The density matrix pf(t ) describes the state of a molecule A, which is formed at the moment t. It is a shorthand description of the fact that, within the model assumed, the state of a molecule is unambiguously determined by the mean density matrices Pj(t ) of all the components of the equilibrium considered and by the probabilities ji, a ... 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

In accordance with this definition the Heisenberg operators ca(t), cjj(f), etc. are equal to the time-independent Schrodinger operators at some initial time to. ca(to) = ca, etc. Density matrix of the system is assumed to be equilibrium at this time p(to) = peq. Usually we can take to = 0 for simplicity, but if we want to use to 0 the transformation to Heisenberg operators should be written as... 

The strong role of collisions in decohering a system is readily seen by considering fie density matrix pj of a system that has reached thermal equilibrium at tempera-rs T through collisional relaxation, that is, pf = Qexp(—Hs./kgT). Here Hs is the system Hamiltonian, Q is a normalization factor, and kB is the Boltzmann constant, fmsiderable insight is obtained if we cast the density matrix in the energy repre-... 

The general question is, if the oscillator is prepared at time t = 0 in a nonequilibrium state with some initial density matrix p(0), how does the oscillator s energy relax back to equilibrium Or to be more precise, defining the nonequilibrium average energy by E(t) = 5]nEnPn(t), where the oscillator s nonequilibrium occupation probabilities are Pn(t), what is the time dependence of E(t) ... 

This is a very important result. It means that at equilibrium, where the phases are truly random, all off-diagonal elements of the density matrix must be zero. The diagonal elements are then given by c 2, which we recognize as the probability of occurrence of state n>, or the population of state n>. 

In Section 11.3 we find that the density matrix for a spin system at equilibrium can be separated into the unit matrix 1 and other terms pertaining to populations of spin states. The unit matrix is unaffected by rf pulses or any other evolution of the density matrix hence it is conventional to delete it, and we do so. Some authors introduce a new symbol for the truncated matrix, but most do not. We continue to use the symbol p for this truncated density matrix. 

Let s now look at the density matrix as it evolves for the pulse sequences that we outlined in Section 11.5. We have illustrated INEPT and related pulse sequences by the two-spin system in which I = H and S = 13C.The equilibrium density matrix can be considered as the sum of two matrices ... 

---