Eigenstate properties

This section begins with a brief description of the basic light-molecule interaction. As already indicated, coherent light pulses excite coherent superpositions of molecular eigenstates, known as wavepackets , and we will give a description of their motion, their coherence properties, and their interplay with the light. Then we will turn to linear and nonlinear spectroscopy, and, finally, to a brief account of coherent control of molecular motion. 

Moreover, because there are only two eigenstates, it follows from the completeness property, the vanishing of (n VQ// n) and the angular momentum commutation relations that... 

Onee a partieular value f] is observed in a measurement of F, this same value will be observed in all subsequent measurements of F as long as the system remains undisturbed by measurements of other properties or by interaetions with external fields. In faet, onee f has been observed, the state of the system beeomes an eigenstate of F (if it already was, it remains unehanged) ... 

Thus, the expansion of / in terms of eigenstates of the property being measured dietated by the fifth postulate above is already aeeomplished. The only two terms in this expansion eorrespond to momenta along the y-axis of 2h/Ly and -2h/Ly the probabilities of observing these two momenta are given by the squares of the expansion eoeffieients of / in terms of the normalized eigenfunetions of -ihd/dy. The funetions (l/Ly)F2 exp(i27iy/Ly) and... 

This distinction between the characteristic eigenstates of the system with their intrinsic properties and the act of preparing the system in some state that may be a superposition of these eigenstates is essential to keep in mind when applying quantum mechanics to experimental observations. 

An important property of the electron Hamiltonian (Eq. (3.3)) is that for arbitrary hopping amplitudes the spectrum of the single-electrons slates is symmetric with respect to c=0 if is the electron amplitude on site n of an eigenstate with energy c, then the state with amplitudes —)"< > is also an eigenstate, with energy -c. In particular, in the uniformly dimerized stale, the gap between the empty conduction and the completely filled valence bands ranges from -A, to A(). 

The von Neumann Projection Operators.—Consider the eigenstates n > in the Hilbert space of N particles with the properties ... 

Papalexi, N., 356,371,377,381 Paramagnetic crystals point groups for, 737 symmetry properties of eigenstates, 745... 

Symmetry of magnetic structures, 726 Symmetry properties of eigenstates of a paramagnetic crystal, 745... 

A comparison of the theory for EOM-CC properties, which empahsize eigenstates and generalized expectation values, and the derivative approach of CCLR has been presented. The usual form of perturbation theory for properties, employ only lower-order wavefunctions in their determination. CCLR involves consideration of wavefunctions of the same order as the energy of interest, but this ensures extensivity of computed properties. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

According to the basic principle of quantum mechanics, any measurable property can be computed ab initio if the total wave function y/ describing the quantum eigenstate of the system is known, since it contains the complete... 

Perturbation theory also provides the natural mathematical framework for developing chemical concepts and explanations. Because the model H(0) corresponds to a simpler physical system that is presumably well understood, we can determine how the properties of the more complex system H evolve term by term from the perturbative corrections in Eq. (1.5a), and thereby elucidate how these properties originate from the terms contained in //(pertJ. For example, Eq. (1.5c) shows that the first-order correction E11 is merely the average (quantum-mechanical expectation value) of the perturbation H(pert) in the unperturbed eigenstate 0), a highly intuitive result. Most physical explanations in quantum mechanics can be traced back to this kind of perturbative reasoning, wherein the connection is drawn from what is well understood to the specific phenomenon of interest. 

By using the closure properties of the eigenstates one obtains easily ... 

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