Common set of eigenfunctions

2 Common set of eigenfunctions Let the square of the angular momentum be introduced 

Because the remaining components do not commute with lz, they have not the same set of eigenstates and thus they cannot be determined simultaneously. Moreover, lz and P both commute with the Hamiltonian H 

Thus we can use an indexation P,) = E, X, /x) for a set of eigenstates which simultaneously are eigenstates of the operators H, P and lz. This result plays a key role in the theory of the electronic structure of atoms one can expand an arbitrary eigenfunction as a sum over eigenfunctions common for the operators H, P and lz 

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Since Jx, Jy, Jz all commute with J2, but not with each other, only one of the components (taken to be /z) has a common set of eigenfunctions with J 2. These eigenfunctions are called uJm or, in Dirac notation, j m) ... 

Perturbations may be divided into two categories on the basis of whether H and Ef have a common set of eigenfunctions. If they do, then only the ground-state term on the right-hand side of Eq. (3.32) is nonzero and the perturbation can be evaluated from ili alone. An example is the case in which H is the nuclear quadrupole moment. The first-order energy 0 if and only if the electric field has a nonzero gradient at... 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hermitian operators, and the mathematical result that only operators which commute have a common set of eigenfunctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to determine the values of the two quantities A and B, and that the corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both refiect the same quantum-mechanical state of the system. If the wavefimction is neither an eigenfiinction of jlnor Ji, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefimction / in terms of the eigenfunctions of the relevant operators... 

We will sometimes use the theorem that, if two linear and Hermitian operators A and B eommute, they have a common set of eigenfunctions and vice versa. 

When the parity operator commutes with the Hamiltonian operator H, we can select a common set of eigenfunctions for these operators, as proved in Section 7.4. The eigenfunctions of H are the stationary-state wave functions i/r,. Hence when... 

In addition to Pz, a symmetric top has a component of the total angular momentum Pa Pz) along the symmetry axis, which is a constant of motion. The quantities K, P, Pz, and Pa commute with each other and hence have a common set of eigenfunctions denoted by f jkm = J, K,M). The matrix elements in the symmetric-top basis,... 

The Hamiltonian H therefore possesses a common set of eigenfunctions with and Sz and we shall assume that the exact wave function (2.4.1) is a spin eigenfunction with quantum numbers S and M, respectively. Consequently, when calculating an approximation to this state, we shall often find it convenient to restrict the optimization to the part of the Fock space that is spanned by spin eigenfunctions with quantum numbers S and M. It is therefore important to examine the spin properties of determinants. 

First, we show that if there exists a common complete set of eigenfunctions for two linear operators then these operators commute. Let A and B denote two linear operators that have a common complete set of eigenfunctions gi,... 

THEOREM 4. If the linear operators A and B have a common complete set of eigenfunctions, then A and B commute. 

Thus, when A and B commute, it is always possible to select a common complete set of eigenfunctions for them. For example, consider the hydrogen atom, where the operators and H were shown to commute. If we desired, we could take the phi factor in the eigenfunctions of H as sin m and cos m4> (Section 6.6). If we did this, we would not have eigenfunctions of except for w = 0. However, the linear combinations... 

Extension of the above proofs to the case of more than two operators shows that for a set of Hermitian operators A,B,C,... there exists a common complete set of eigenfunctions if and only if every operator commutes with every other operator. 

THEOREM 5. If the Hermitian operators A and B conunute, we can select a common complete set of eigenfunctions for than. 

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. 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

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