Operators eigenfunctions

Operators, Eigenfunctions and Eigenvalues. Inasmuch as all matter is (from a chemist s point of view) a combination of atomic nuclei and electrons, it is clear that the motions, or rather the distributions, of these particles must explain all observable properties, and there are many of these for example the energy of a molecule, its dipole moment, and its shape. In the end the observable property must be a number (with the relevant units) and these values are called eigenvalues . How can one obtain them ... 

In Section 3 we provided an example for the utilization of the new basis. Ihe new polynomials were effective in the operator projection methods used there. We expect similar polynomials be equally effective when employed in other spectral methods. Ihe conputational econorty achieved by capturing the correct asymptotic behavior of the operator eigenfunctions and still retaining orthogonality will be especially useful in the solution of multidimensional problems. 

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

This provides a recipe for calculating the average value of the system property associated with the quantum-mechanical operator A, for a specific but arbitrary choice of the wavefiinction T, notably those choices which are not eigenfunctions of A. 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

In a more favourable case, the wavefiinction ]i might indeed correspond to an eigenfiinction of one of the operators. If = //, then a measurement of A necessarily yields and this is an unambiguous result. Wliat can be said about the measurement of B in this case It has already been said that the eigenfiinctions of two commuting operators are identical, but here the pertinent issue concerns eigenfunctions of two operators that do not conmuite. Suppose / is an eigenfiinction of A. Then, it must be true that... 

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts these functions are called wavefunetions... 

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. 

Because the property corresponding to the operator is about to be measured, we express the above / in terms of the eigenfunctions ofL ... 

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... 

We therefore conclude that the act of carrying out an experimental measurement disturbs the system in that it causes the system s wavefunction to become an eigenfunction of the operator whose property is measured. If two properties whose corresponding operators commute are measured, the measurement of the second property does not destroy knowledge of the first property s value gained in the first measurement. 

On the other hand, as detailed further in Appendix C, if the two properties (F and G) do not commute, the second measurement destroys knowledge of the first property s value. After the first measurement, P is an eigenfunction of F after the second measurement, it becomes an eigenfunction of G. If the two non-commuting operators properties are measured in the opposite order, the wavefunction first is an eigenfunction of G, and subsequently becomes an eigenfunction of F. 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

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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