Intrinsic angular momentum

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 Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

The last term is the intrinsic change in the operator P, which is zero when P does not formally depend on t, as is the case for momentum, angular momentum, and spin operators. In deriving Eq. (7-74), no use was made of the adiabatic property of the w-functions. Therefore, it holds for all time-dependent bases. In the moving representation, w = , D = H by virtue of Eq. (7-49), and (7-74) reverts to Eq. (7-59). 

Analysis of the prototypical resonant swing spring model [11-13] shows that Fermi resonance with conserved angular momentum is an intrinsically three-dimensional phenomenon. The form of the 3x3 monodromy matrix was given. 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

G. E. Uhlenbeck and S. Goudsmit (1925) explained the splitting of atomic spectral lines by postulating that the electron possesses an intrinsic angular momentum, which is called spin. The component of the spin angular momen-... 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

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. 

A spinning electron also has a spin quantum number that is expressed as 1/2 in units of ti. However, that quantum number does not arise from the solution of a differential equation in Schrodinger s solution of the hydrogen atom problem. It arises because, like other fundamental particles, the electron has an intrinsic spin that is half integer in units of ti, the quantum of angular momentum. As a result, four quantum numbers are required to completely specify the state of the electron in an atom. The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers. We will illustrate this principle later. 

In his last paper Dunham obtained a formula for values of spectral terms for a particular isotopic species i in a particular electronic state, which we suppose generally to be of symmetry class 2 or 0 implying neither net electronic orbital nor net intrinsic electronic angular momentum ... 

The spin quantum number (/Mj) gives the projection of the intrinsic angular momentum of the electron, with values + and —Two oppositely oriented spins on the same orbital are graphically represented by arrows in opposite directions (Ti). 

Both protons and neutrons in the nucleus have an intrinsic angular momentum resulting from spinning of nucleons,... 

---