Commuting observables

In (10-124) and (10-125), n and m refer to the eigenvalues of a complete set of commuting observables so Snm stands for a delta function m those observables in the set that have a continuous spectrum, and a Kronecker 8 in those that have a discrete spectrum. 

Since the operators P commute with one another we can choose a representation in which every basis vector is an eigenfunction of all the P s with eigenvalue It should be noted that the specification of the energy and momentum of a state vector does not uniquely characterize the state. The energy-momentum operators are merely four operators of a complete set of commuting observables. We shall denote by afi the other eigenvalues necessary to specify the state. Thus... 

States can be simultaneously eigenstates of more than one observable. In this case the observables commute. If the simultaneous eigenstates of a set of commuting observables form a complete set they span a new space which is the direct product space of the spaces spanned by the eigenstates of each of the observables in the set. The dimension of the new space is the product of the dimensions of the spaces spanned by the eigenstates of the individual observables. 

The set of commuting observables can be considered as a new observable. In this way we can extend the space associated with a particular dynamical property of a system to a more general dynamical property of that system or to dynamical properties of a larger system that includes it. 

Since L and Lz are commuting observables they have simultaneous eigenstates m), which obey the eigenvalue equations... 

The eigenvectors IJm) of and Jz are elements of the product space of simultaneous eigenvectors of the commuting observables The 2ji + l)(2j2 + l)-dimensional product space is partitioned into subspaces of dimension 2j + I, where... 

If two operators commute, there is no restriction on the accuracy of their simultaneous measurement. For example, the x- and y-coordinates of a particle can be known at the same time. An important theorem states that two commuting observables can have simultaneous eigenfunctions. To prove this, write the eigenvalue equation for an operator A ... 

In the description of nature afforded by quantum mechanics, one classifies and characterizes the state of a total system in terms of the eigenvalues of a set of commuting observables acting on an element of the Hilbert space, the state vector. Molecular orbital theory in its canonical representation as originally... 

An important example of a maximal set of commuting observables with a continuous spectrum of eigenvalues is provided by the operators 4r representing the position coordinates of a set of particles. The state function in terms of the coordinate representation is given by... 

Assuming that the parameters of the system are not altered, the variation of the action integral in eqn (8.78) arises only from infinitesimal changes of the sets of commuting observables at the two times t and tj. We have previously seen (eqn (8.41)) that such transformations may be characterized in terms of the generators of infinitesimal unitary transformations jF(t,) and F(t2), which act on the eigenvectors and q,2> 2>- Comparing eqns (8.41) and... 

Corollary V.l) to any complete set of commuting observables (CSCO) there corresponds a complete set of commuting effective observables CSCEO) if the mappings are norm-preserving. This corollary is proven using the definition of a complete commuting set. 

Since the operators corresponding to all the components of the positions of the particles in the system commute, the coordinates form a complete set of commuting observables. Consequently any ket xfr) can be expanded in terms of eigenkets of the complete set, i.e. 

For separable systems the Schrodinger equation, represented by a partial differential equation, is mapped onto uni-dimensional differential equations. The eigenvalues (separation constants) of each of the 3 uni-dimensional (in general n uni-dimensional) differential equations can be used to label the eigenfunctions (r) and hence serve as quantum numbers. Integrability of a n-dimensional Hamiltonian system requires the existence of n commuting observables O/, 1 i in involution ... 

The stationary states, H x> = E x> includes any other quantum numbers comprising a complete set of commuting observables), determine a basis. We assume that the Hamiltonian has a continuous spectrum only, or at least that the discrete states of H are orthogonal to ho. Moreover, we assume that the spectrum is bounded from below E < E < oo. The complefeness relation... 

If there are other variables whose operators commute with A, measurement of enough of these variables can put the system into a known state. We say that such a set of variables is a complete set of commuting observables. For example, assume that the operators A and B commute and that they have a set of common eigenfunctions fn, fn,---, fii, such that... 

If the variables A and B constitute a complete set of commuting observables the system is no w in the state corresponding to the wave function fkm If there are other commuting observables in the complete set, additional measurements must be made. We will see an important example of this principle in Chapter 17, when we will find that four variables constitute a complete set of commuting observables for the electronic motion of a hydrogen atom. 

The measurement on the same system of a complete set of commuting observables suffices to put the system into a state that is completely known, even though only partial information is available about the state of the system prior to the measurements. 

There is nothing unique about the z direction. One could choose Lx or Ty as a member of a set of commuting observables instead of L. In that event, the functions would be different, and would correspond to cones in Figure 17.4 that would be oriented around either the x axis or the y axis. We choose to emphasize L since its operator is simpler in spherical polar coordinates than those of the other components. 

We are now able to apply the concept of a complete set of commuting observables to the hydrogen atom. As explained in Chapter 16, measurement of a complete set of... 

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