Operators, angular momenta electron spin

The magnetic properties of a molecule or ion result primarily from the interaction of the angular momentum and spin of the electrons and nuclei with the magnetic field. To deal with any quantum-mechanical theory of magnetic properties, we need to know some basic properties of the angular-momentum operators and their eigenfunctions. 

In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. 

The operators for the electron spin angular momentum obey the following relations, which are similar to those shown in Eqs. (1-32) and (1-33) ... 

For light atoms and certain linear radicals, s and 1 combine via the Russell-Saunders Coupling scheme to give a total angular momentum, j. EPR spectroscopy then interrogates transitions between the different j levels. Hence, since the Hamiltonian operator for an electron spin in a molecule contains a term representing the effect of... 

We will discuss the Dirac equation for the one-electron atom in more detail in chapter 7. Here we are only interested in the symmetry properties of the Hamiltonian for such systems. We know that for the corresponding nonrelativistic case, angular momentum and spin are normal constants of motion, represented by operators that commute with the Hamiltonian. In particular... 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

A particle possesses an intrinsic angular momentum S and an associated magnetic moment Mg. This spin angular momentum is represented by a hermitian operator S which obeys the relation S X S = i S. Each type of partiele has a fixed spin quantum number or spin s from the set of values 5 = 0, i, 1,, 2,. .. The spin s for the electron, the proton, or the neutron has a value The spin magnetie moment for the electron is given by Mg = —eS/ nie. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The singlet function corresponds to zero total electron spin angular momentum, S = 0 the triplet functions correspond to S = 1. Operating on these functions with the spin Hamiltonian, we get ... 

Here, L is the angular momentum operator, ge is the g value for the free electron (ge = 2.0023), and X is the spin-orbit coupling constant. Considering only the z direction, this equation becomes... 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

---