Operators, angular momenta Hermitian

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. 

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

On the other hand, the entity conserved in a closed system due to the isotropy of space is the orbital angular momentum of the system. Apart from a constant factor, the operator fa x Va must therefore correspond to the orbital angular momentum. Further, the angular momentum is an observable (i.e., real valued). Thus the corresponding operator ought to be Hermitian. An operator is said to be Hermitian if it obeys the turn-over rule, that is,... 

The operator F must be taken to be Hermitian (to ensure that the transformation (7.246) is unitary), totally symmetric and of odd degree in the angular momenta. The last requirement follows from the fact that the non-vanishing commutation relationships between angular momentum components, say. / , reduce the power of the operators by one. The result of the transformation is therefore of even power in / which is required if the term in %nv is to be symmetric with respect to time reversal. To illustrate the procedure, let us consider the operator chosen by Brown and Watson [17] ... 

Further, the orbital angular momentum is a purely imaginary Hermitian operator (for details, see, e.g., [121, 122]). Thus - and as explained in detail in the Appendix - another rule results ... 

The operator serves to extract information from the state vectors and may correspond to any physical observable such as position, momentum, angular momentum (spin or orbital) or energy. The state vector itself is not observable. In most systems the number of eigenfunctions is infinite and it is an axiom of quantum mechanics that the set of all eigenfunctions of a Hermitian operator forms a complete set. These eigenfunctions define a Hilbert space on which the operator acts. 

It is well known that the angular momentum of a quantum mechanical system is specified by a representation of the SU(2) algebra. If the corresponding enveloping algebra contains a uniquely defined scalar (the Casimir operator), the polar decomposition of the angular momentum can be obtained [51]. This polar decomposition determines a dual representation of the SU(2) algebra expressed in terms of so-called phase states [51], In particular, the Hermitian operator of the SU(2) quantum phase can be constructed [51],... 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p Py,Pz by the appropriate operators. Hie inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

Analogous to the orbital angular-momentum operators L, L L L, we have the spin angular-momentum operators S, S, Sy, S which are postulated to be linear and Hermitian. is the operator for the square of the magnitude of the total spin angular momentum of a particle. is the operator for the z component of the particle s spin angular momentum. We have... 

In quantum mechanics, -i9t corresponds to the angular momentum operator so that eqn (8.10) is equivalent to the Hermitian property of this operator.)... 

Each of these four operators is Hermitian. By Theorem 8.3, these commutation relations mean that there are functions that are simultaneously eigenfunctions of and of any one component of the angular momentum however, because the component operators do not commute with each other, a mutually commuting set of operators including and two (or three) component operators cannot be formed. 

---