Simultaneous eigenfunctions

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. 

In the first experiment, the fact that they are mutually commutative allowed us to expand the 64 % probable eigenstate with L=1 in terms of functions that were eigenfunctions of the operator for which measurement was about to be made without destroying our knowledge of the value of L. That is, because and can have simultaneous eigenfunctions, the L = 1 function can be expanded in terms of functions that are eigenfunctions of both and L. This in turn, allowed us to find experimentally the... 

Since and commute you would expect 0,0> to be simultaneous eigenfunctions of both. 

Lx and do not commute. It is unexpected to find a simultaneous eigenfunction ( 0,0>) of both. .. for sure these operators do not have the same full set of eigenfunctions. 

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

Since commuting operators have simultaneous eigenfunctions, it follows that correct eigenfunctions of the BO Hamiltonian must also be eigenfunctions of and Sz with eigenvalues S (S + 1) andM5 = S,S 1,., —S. All 2S + 1 members of a... 

Suppose the members of a complete set of functions tpi are simultaneously eigenfunctions of two hermitian operators A and B with eigenvalues a,- and j3i, respectively... 

Thus, the functions tpi are eigenfunctions of the commutator A, S] with eigenvalues equal to zero. An operator that gives zero when applied to any member of a complete set of functions is itself zero, so that A and B commute. We have just shown that if the operators A and B have a complete set of simultaneous eigenfunctions, then A and B commute. 

We now prove the converse, namely, that eigenfunctions of commuting operators can always be constructed to be simultaneous eigenfunctions. Suppose that Afi = atfi and that [A, 5] = 0. Since A and B commute, we have... 

This analysis can be extended to three or more operators. If three operators A, B, and C have a complete set of simultaneous eigenfunctions, then the argument above shows that A and B commute, B and C eommute, and A and C commute. Furthermore, the converse is also true. If A eommutes with both B and C, and B commutes with C, then the three operators possess simultaneous eigenfunctions. To show this, suppose that the three operators commute with one another. We know that since A and B commute, they possess simultaneous eigenfunctions such that... 

We note here that if A commutes with B and B commutes with C, but A does not eommute with C, then A and B possess simultaneous eigenfunctions, B and C possess simultaneous eigenfunetions, but A and C do not. The set of simultaneous eigenfunetions of A and B will differ from the set for B and C. An example of this situation is diseussed in Chapter 5. 

We have shown in Section 3.5 that commuting hermitian operators have simultaneous eigenfunctions and, therefore, that the physical quantities associated with those operators can be observed simultaneously. On the other hand, if the hermitian operators A and B do not commute, then the physical observables A and B cannot both be precisely determined at the same time. We begin by demonstrating this conclusion. 

Some or all of the eigenvalues may be degenerate, but each eigenfunction has a unique index i. Suppose further that the system is in state aj), one of the eigenstates of A. If we measure the physical observable A, we obtain the result aj. What happens if we simultaneously measure the physical observable B To answer this question we need to calculate the expectation value (B) for this system... 

We are now ready to obtain the set of simultaneous eigenfunctions for the commuting operators N and H. The ground-state eigenfunction 0) has already been determined and is given by equation (4.31). The series of eigenfunctions 11), 2),... are obtained from equations (4.34b) and (4.18b), which give... 

We now apply the results of the quantum-mechanical treatment of generalized angular momentum to the case of orbital angular momentum. The orbital angular momentum operator L, defined in Section 5.1, is identified with the operator J of Section 5.2. Likewise, the operators I , L, Ly, and are identified with J, Jx, Jy, and Jz, respectively. The parameter j of Section 5.2 is denoted by I when applied to orbital angular momentum. The simultaneous eigenfunctions of P and are denoted by Im), so that we have... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

Since the variable r does not appear in any of these operators, their eigenfunctions are independent of r and are functions only of the variables 6 and tp. The simultaneous eigenfunctions Im) of I and L will now be denoted by the function YimiO, tp) so as to acknowledge explicitly their dependence on the angles 6 and tp. 

We have shown that the simultaneous eigenfunctions 0, tp) of the operators H, I , and Lz have the form... 

There are only two functions which are simultaneous eigenfunctions of H(l, 2) and > with respective eigenvalues E and 1. These functions are the combinations... 

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