Eigenfunctions commuting operators have simultaneous

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... 

Section 6-11 Proof That Commuting Operators Have Simultaneous 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... 

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. 

The operator in square brackets on the left-hand side of equation (10.33) commutes with the operator P and with the operator in (5.31c), because P commutes with itself as well as with and neither P nor contain the variable R. Consequently, the three operators have simultaneous eigenfunctions. From the argument presented in Section 6.2, the nuclear wave function Xv R, 0, cp) has the form... 

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... 

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... 

The second line of this equation follows from (7.102) above. We note that the awkward sin 0 factors in (7.89) have now disappeared. As Hougen points out, the eigenfunctions of the true Hamiltonian involve one less variable and so one less quantum number than the eigenfunctions of the artificial Hamiltonian and consequently the two operators cannot be completely isomorphic. However, a simple restriction on the extra quantum number in the artificial problem identifies that part of the full artifical Hamiltonian which is isomorphic with the true operator. Since the isomorphic Hamiltonian commutes with (Jz — W-), the two operators have a set of simultaneous eigenfunctions. Equation (7.102) states that only those eigenfunctions of the isomorphic Hamiltonian which have an eigenvalue of zero for (Jz - Wz) are eigenfunctions of the true Hamiltonian. 

It may be that the wave functions are eigenfunctions of two non-commuting operators corresponding to physical quantities such as p (momentum) and q (position) respectively. Then, by measuring either A or B in system I, it becomes possible to predict with certainty and without disturbing the second system, either the value of Pk or qr. In the first case p is an element of reality and in the second case q is an element of reality. But ipk and commuting operators cannot have simultaneous reality. It was inferred that quantum theory is incomplete. 

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 ... 

At this point the reader may feel that we have done little in the way of explaining molecular symmetry. All we have done is to state basic results, normally treated in introductory courses on quantum mechanics, coimected with the fact that it is possible to find a complete set of simultaneous eigenfunctions for two or more commuting operators. However, as we shall see in section A 1.4.3,2. the fact that the molecular Hamiltonian //commutes with and is intimately connected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing through the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the ... 

The atomic Hamiltonian (t of (11.1) (which omits spin-orbit interaction) does not involve spin and therefore commutes with the total-spin operators 5 and S. The fact that commutes with ft is not enough to show that the atomic wave functions are eigenfunctions of 5 The Pauli antisymmetry principle requires that each tp must be an eigenfunction of the exchange operator with eigenvalue —1 (Section 10.3). Hence must also commute with if we are to have simultaneous eigenfunctions of H, S, and ic. Problem 11.16 shows that [. , 4fc] = 0, so the atomic wave functions are eigenfunctions of We have = S S + l)feV each atomic state can be characterized by a total-electronic-spin quantum number S. 

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. 

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. 

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