Commutation rule

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

From Eq. (7-la), one may construct Lx and Ly by cyclical change of coordinates, and the three operators are found to satisfy the commutation rules... 

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

The proof of the theorem affirming that J8 is a proper quantum mechanical angular momentum involves only an expansion of (Ji + J2) x (Ji + J2) with subsequent use of the commutation rules for Jj and J2, and the fact that Jj and J2 commute because they act in... 

The operators 0 = Qi>Qa>Qs) and P = (P P P3) are conjugate to each other in the sense that they obey the following commutation rule ... 

Stated more abstractly, in quantum mechanics, a particle is characterized by a set of dynamical variables, p,q, which are represented by operators that obey the fundamental commutation rules... 

For a free noninteracting spinning particle, invariance with respect to translations and rotations in three dimensional space, i.e., invariance under the inhomogeneous euclidean group, requires that the momenta pl and the total angular momenta J1 obey the following commutation rules... 

These commutation rules imply that if the state is an eigenstate of P . with eigenvalue p ... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

The commutation rules of (+ x) and are easily derived from those of On,Ok- They are ... 

The above commutation rules suggest that an explicit representation of Q is given by... 

These operators satisfy the following commutation rules... 

All other products of y-matrices can, by using the commutation rules, be reduced to one of these sixteen elements. The proof of their linear independence is based upon the fact that the trace of any of these matrices except for the unit matrix, I, is zero. If Tr is any one of these matrices, then rrr, generates again one of the T s, the unit matrix... 

Since the matrices — yu,yt , "/lT, and y" all obey the same commutation rules as -/ (take the hermitian adjoint, transpose and complex conjugate of Eq. (9-254) ) it follows from theorem G, that there exist nonsingular matrices A, B, C, D, such that... 

Although the previous discussion assumed no properties of the y-matrices beyond their commutation rules, in most of our subsequent discussion we shall specialize to a particular representation of these matrices, namely the one in which y° is-hermitian and the y are anti-hermitian. In this representation A = y° and u — u y°. 

Actually there exists in the FW representation another polarization operator, Sf, which also satisfies the commutation rules... 

The normalization requirement of the one and many particle states is satisfied if we impose the following commutation rules on the b and d... 

The s, therefore, satisfy angular momentum commutation rules. Since each of these matrices has eigenvalues 1 and 0, they form a representation of the angular momentum operators for spin 1. 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

We postulate the equal time commutation rules for the field operators to be... 

Similarly, the commutation rules (9-587) and (9-588) allow us to deduce that... 

In a manner similar to the above, one verifies that the commutation rules (9-587) and (9-588) guarantee that P as given by Eq. (9-581) is the generator for an infinitesimal spatial translation in the sense that Eq. (9-583) is satisfied for the field operators. Similarly, one verifies that the three components of the vector operator... 

We have now achieved one objective, that of expressing the hamiltonian in essentially diagonal form. In order to deal with operators that satisfy S-function commutation rules rather than SJy commutation rules, where... 

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