Quantum spin states

The measurable component of electron spin has the magnitude of h/2, which produces two allowed quantum spin states, ms = +Vi (designated by convention as a > spin), and ms = -V2 (or p > spin). When a free electron is placed in a strong magnetic field, its magnetic moment ps behaves like a tiny bar magnet. It will line up either parallel or antiparallel to the field H. The energy of this interaction is... 

With the aim of quantitatively predicting the orientational order of rigid solutes of small dimensions dissolved in the nematic liquid crystal solvent, 4-n-pentyl-4 cyanobiphenyl (5CB), an atomistic molecular dynamics (MD) computer simulation has been applied. It is found that for the cases examined the alignment mechanism is dominated by steric and van der Waals dispersive forces. A computer simulation of the deuterium NMR spectra of molecules in a thin nematic cell has been carried out and the director distribution in the cell has been studied. An experiment for the direct estimation of an element of the order matrix from H NMR spectra of strongly dipolar coupled spins that is based on the multiple quantum spin state selected detection of single quantum transitions has been proposed. The experiment also enables obtaining nearly accurate starting dipolar... 

The negative sign in equation (b 1.15.26) implies that, unlike the case for electron spins, states with larger magnetic quantum number have smaller energy for g O. In contrast to the g-value in EPR experiments, g is an inlierent property of the nucleus. NMR resonances are not easily detected in paramagnetic systems because of sensitivity problems and increased linewidths caused by the presence of unpaired electron spins. 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

The electrons do not undergo spin inversion at the instant of excitation. Inversion is forbidden by quantum-mechanical selection rules, which require that there be conservation of spin during the excitation process. Although a subsequent spin-state change may occur, it is a separate step from excitation. 

According to the law for the eigenvalues, our determinant could possibly be a mixture of spin states associated with the quantum numbers S = m, m- -1,. . N. However, in the special case... 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

As shown in Section II.D(2), the determinant of Eq. III.133 can be brought to correspond to a pure spin state by imposing a certain condition (11.61) on the relation between p+ and p. which corresponds to the pairing of the electrons. If p+ and p are permitted to vary independently of each other, the determinant is no longer a pure spin state but a mixture of states associated with the quantum numbers... 

According to quantum mechanics, an electron has two spin states, represented by the arrows T (up) and l(down) or the Greek letters a (alpha) and P (beta) We can think of an electron as being able to spin counterclockwise at a certain rate... 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

Coherence A condition in which nuclei precess with a given phase relationship and can exchange spin states via transitions between two eigenstates. Coherence may be zero-quantum, single-quantum, double-quantum, etc., depending on the AM of the transition corresponding to the coherence. Only single-quantum coherence can be detected directly. 

It is thus evident that the experimental results considered in sect. 4 above are fully consistent with the interpretation based on absolute reaction rate theory. Alternatively, consistency is equally well established with the quantum mechanical treatment of Buhks et al. [117] which will be considered in Sect. 6. This treatment considers the spin-state conversion in terms of a radiationless non-adiabatic multiphonon process. Both approaches imply that the predominant geometric changes associated with the spin-state conversion involve a radial compression of the metal-ligand bonds (for the HS -> LS transformation). 

Fig. 35. Spin-state relaxation rate constant k versus temperature T for PSS-doped [Fe(6-Mepy)2(py)tren](CIOj2- Experimental data are indicated by filled circles. The solid line represents the fit to the tuimeling model of Hopfield, the dashed line the fit to the quantum mechanical theory of Buhks et al. According to Ref [138]...

A quantum-mechanical description of spin-state equilibria has been proposed on the basis of a radiationless nonadiabatic multiphonon process [117]. Calculated rate constants of, e.g., k 10 s for iron(II) and iron(III) are in reasonable agreement with the observed values between 10 and 10 s . Here again the quantity of largest influence is the metal-ligand bond length change AR and the consequent variation of stretching vibrations. 

The state of a particle with zero spin s = 0) may be represented by a state function (r, t) of the spatial coordinates r and the time t. However, the state of a particle having spin 5 (5 7 0) must also depend on some spin variable. We select for this spin variable the component of the spin angular momentum along the z-axis and use the quantum number ms to designate the state. Thus, for a particle in a specific spin state, the state function is denoted by (r, ms, t), where ms has only the (2s + 1) possible values —sh, (—s + )h,... 

The NMR signal arises from a quantum mechanical property of nuclei called spin . In the text here, we will use the example of the hydrogen nucleus (proton) as this is the nucleus that we will be dealing with mostly. Protons have a spin quantum number of V . In this case, when they are placed in a magnetic field, there are two possible spin states that the nucleus can adopt and there is an energy difference between them (Figure 1.1). 

Levitt et al.69 have used the double quantum solid state NMR in the studies of bond lengths for a series of five 13C labelled samples of rhodopsin. On the basis of DQ-filtered signal trajectories and numerical spin simulations of the signal points, the through-space dipole-dipole coupling between neighbouring 13C nuclei has been estimated. Estimated dipole-dipole couplings have been converted into the intemuclear distances (Table 2) [32],... 

The electron spin resonance spectrum of a free radical or coordination complex with one unpaired electron is the simplest of all forms of spectroscopy. The degeneracy of the electron spin states characterized by the quantum number, ms = 1/2, is lifted by the application of a magnetic field, and transitions between the spin levels are induced by radiation of the appropriate frequency (Figure 1.1). If unpaired electrons in radicals were indistinguishable from free electrons, the only information content of an ESR spectrum would be the integrated intensity, proportional to the radical concentration. Fortunately, an unpaired electron interacts with its environment, and the details of ESR spectra depend on the nature of those interactions. The arrow in Figure 1.1 shows the transitions induced by 0.315 cm-1 radiation. 

Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

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