Wavefunction eigenfunction

A. Express the wavefunction (eigenfunction) as the sum of orthogonal, normalized wavefunctions typically the latter would be spin functions denoted by pj... 

In the four examples given below, a particle (or electron) is allowed to move freely. The only difference is the shape of the box, in which the particle travels. As will be seen later, different shapes give rise to different boundary conditions, which in turn lead to different allowed energies (eigenvalues) and wavefunctions (eigenfunctions). 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

Hartree-Fock wavefunction, is an eigenfunction of and the corresponding oth-order energy Eg° is equal to the sum or orbital energies for the occupied molecular... 

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. 

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. 

This wavefunction needs to be expanded in terms of the eigenfunctions of the angular... 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

The resulting wavefunction is not necessarily an eigenfunction of the spin operator S. This may or may not matter, depending on the application. 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

The initial wavefunction is a product of a specific rovibrational eigenfunction for the reactants and a localized translational wavepacket for R ... 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

The concept of scarred quantum wavefunctions was introduced by Eric Heller (E.J. Heller, 1984) a little over 20 years ago in work that contradicted what appeared at the time to be a reasonable expectation. It had been conjectured (M.V. Berry, 1981) that a semiclassical eigenstate (when appropriately transformed) is concentrated on the region explored by a generic classical orbit over infinite times. Applied to classically chaotic systems, where a typical orbit was expected to uniformly cover the energetically allowed region, the corresponding typical eigenfunction was anticipated to be a superposition of plane... 

The extra h was introduced to make the exponent dimensionless. The wavefunctions Xi 3X6 eigenfunctions of Hi with eigenvalues Et, the Xf eigenfunctions of Hf. Hence ... 

Spin-restricted procedures, signified by an R prefix (e.g. RHF, RMP), constrain the a and (3 orbitals to be the same. As such, the resulting wavefunctions are eigenfunctions of the spin-squared operator (S2) that correspond to pure spin states (doublets, triplets, etc). The disadvantage of this approach is that it restricts the flexibility in the... 

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