Angular spin momentum

As mentioned earlier, we cannot make use of the correspondence principle to derive quantum mechanical spin operators, because spin has no classical analog. Instead, the spin eigenfunctions sms) may be identified with u 1) 

Obviously, the spin eigenfunction sms) is not a function of the spatial coordinates mathematically it is known as a spinor. Different notations are 

The symbols a and p are the ones most familiar to chemists. For the definition of spin operators, it is convenient to utilize the representation of the spin eigenfunctions as the orthonormal basis vectors of a two-dimensional (2D) vector space. In this representation, the spin operators may be written as matrices 

For later convenience, we also define the irreducible tensor operators 

Pauli introduced slightly different spin operators known as the Pauli spin matrices. They are defined by 

In addition to the usual terms of kinetic energy and potential energy, quantum mechanical particles possess another property called spin, which has the dimensions of angular momentum. The values of spin are quantized to half integer or integer multiples ofB. In the following we omit the factor of ft when we discuss spin values, for brevity. If the total spin of a particle is s then there are 2s -k 1 states associated with it, because the projection of the spin onto a particular axis can have that many possible values, ranging from +s to -s in increments of 1. The axis of spin projection is usually labeled the z axis, so a spin of s = 1/2 can have projections on the z axis = +1 /2 and = -1/2 sl spin of s = 1 can have = -1,0, +1, and so on. 

The spin of quantum particles determines their statistics. Particles with half-integer spin are called fermions and obey the Pauli exclusion principle, that is, no two of them can be 

There is a jump in the sequential occupation of states of the Coulomb potential, namely we pass from atomic number 18 with the n = 3, / = 0 and / = 1 states filled to atomic number 21, in which the n = 3, / = 2 states begin to be filled. The reason for this jump has to do with the fact that each electron in an atom does not experience only the pure Coulomb potential of the nucleus, but a more complex potential which is also due to the presence of all the other electrons. For this reason the true states in the atoms are somewhat different than the states of the pure Coulomb potential, which is what makes the states with n = 4 and Z = 0 start filling before the states with n = 3 and Z = 2 the n = 4, Z = 0, m = 0 states correspond to the elements K and Ca, with atomic numbers 19 and 20. Overall, however, the sequence of states in real atoms is remarkably close to what would be expected from the pure Coulomb potential. The same pattern is followed for states with higher n and Z values. 

The other type of particles, with integer spin values, do not obey the Pauli exclusion principle. They are called bosons and obey different statistics than fermions. For instance, under the proper conditions, all bosons can collapse into a single quantum mechanical state, a phenomenon referred to as Bose-Einstein condensation, that has been observed experimentally. 

The same pattern applies to higher spin values. 

The postulates of quantum mechanics discussed in Section 3.7 are incomplete. In order to explain certain experimental observations, Uhlenbeck and Goudsmit introduced the concept of spin angular momentum for the electron. This concept is not contained in our previous set of postulates an additional postulate is needed. Further, there is no reason why the property of spin should be confined to the electron. As it turns out, other particles possess an intrinsic angular momentum as well. Accordingly, we now add a sixth postulate to the previous list of quantum principles. 

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. 

As noted in the previous section, spin is a purely quantum-mechanical concept there is no classical-mechanical analog. 

The spin magnetic moment Mg of an electron is proportional to the spin angular momentum S, 

The hermitian spin operator S associated with the spin angular momentum S has components Sx, Sy, S, so that 

We have obtained a detailed description of the spatial wavefunction, one-electron atoms, equation 

The spin operator acts on spin wavefunctions denoted by xCm ) in a special space which is quite separate from the ordinary three-dimensional coordinate space in which the orbital angular momentum operates. 

Similar equations which can be obtained by cyclic permutation are conveniently summarized by 

These are known as the commutation relations for orbital angular momentum. 

By purely algebraic methods it can be shown that the whole quantum theory of angular momentum follows from these equations. In particular, if 2 is a Hermitian operator which is defined to obey the commutation relations 

---

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... 

The simplest case arises when the electronic motion can be considered in temis of just one electron for example, in hydrogen or alkali metal atoms. That electron will have various values of orbital angular momentum described by a quantum number /. It also has a spin angular momentum described by a spin quantum number s of d, and a total angular momentum which is the vector sum of orbital and spin parts with... 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

Raoult s law) Activity (referenced to Axial spin angular momentum cr, X... 

Electron spin and nuclear spin angular momentum... 

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

For an electron having orbital and spin angular momentum there is a quantum number j associated with the total (orbital + spin) angular momentum which is a vector quantity whose magnitude is given by... 

If the nucleus possesses a spin angular momentum, these states are further split and therefore, perhaps, should not have been called states in the first place However, the splitting due to nuclear spin is small and it is normal to refer to nuclear spin components of states. 

The component of the total (orbital plus electron spin) angular momentum along the intemuclear axis is Qfi, shown in Figure 7.16(a), where the quantum number Q is given by... 

The fourth quantum number is called the spin angular momentum quantum number for historical reasons. In relativistic (four-dimensional) quantum mechanics this quantum number is associated with the property of symmetry of the wave function and it can take on one of two values designated as -t-i and — j, or simply a and All electrons in atoms can be described by means of these four quantum numbers and, as first enumerated by W. Pauli in his Exclusion Principle (1926), each electron in an atom must have a unique set of the four quantum numbers. 

Spin-drehimpuls, m. spin angular momentum, -glied, n. (Math.) spin term, -momentdichte, /. spin momentum density. 

Suppose a quantum system consists of a pair of spin- particles, and that the system is prepared in such a way that (1) the total spin angular momentum is zero (i.e. the... 

This is the property we now call spin angular momentum. Pauli found that he could obtain Stoner s classification of electronic configurations from the following simple assumption which constitutes the famous exclusion principle in its original form. 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

Here L, S, and J are the quantum numbers corresponding to the total orbital angular momentum of the electrons, the total spin angular momentum, and the resultant of these two. Hund predicted values of L, S, and J for the normal states of the rare-earth ions from spectroscopic rules, and calculated -values for them which are in generally excellent agreement with the experimental data for both aqueous solutions and solid salts.39 In case that the interaction between L and S is small, so that the multiplet separation corresponding to various values of J is small compared with kT, Van Vleck s formula38... 

It can also be seen from Fig. 6 that if the T+x or T x states mixed with S, this would involve concomitant electron and nuclear spin flipping in order that the total spin angular momentum be conserved, and this would ultimately produce the same polarization in c- and e-products. This point will be discussed further in Section IV. 

On the right side of Fig. 3-9 are represented the relative energies of the two Tig terms, the and 2g. The ground term is the from the 2g configuration. Spin-allowed electronic transitions (those between terms of the same spin angular momentum - but see also Sections 3.6, 3.7 and Chapter 4) now take place upon excitation from -> A2g, The d-d spectra of octahedrally... 

---