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Density Matrix Equations
Published in Pradip Narayan Ghosh, Laser Physics and Spectroscopy, 2018
The quantum mechanical treatment of atom describes a single atom. Hence, the atom field interaction discussed in the last chapter describes the interaction of a single atom with a classical electromagnetic field. In any experiment, under usual circumstances, we have a large number of atoms interacting with the field. The observed result is an average over the effect of interaction of the electromagnetic field with all these atoms. The atoms may not all be in the same quantum mechanical state. For the atom in one quantum mechanical state, the observed value is an expectation value which is an average over a large number of measurements of the same physical variable in the same state. The atom is described by a set of eigenstates φn. At any instant the general wave function of an atom is () |ψ(t)〉=∑ncn(t)|ϕn〉.
Heisenberg's Matrix Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
An expectation value 〈A〉 of the observable A is the mean value of a series of the specific measurements of the quantum state vector. The quantum state vector provides a probability distribution for the outcomes of each measurement of a particle. There is no state which is simultaneously an eigenstate for all observables.
Pre-Born–Oppenheimer molecular structure theory
Published in Molecular Physics, 2019
Observables in quantum mechanics are computed as the expectation value of the appropriate operator with the wave function of the system. It is straightforward to calculate expectation values of structural parameters with the (all-particle) molecular wave function. However, it is important to recognise that if we calculated the carbon nucleus-proton distance in an organic molecule, we would obtain a single value [98–100] due to the quantum mechanical indistinguishability of identical particles. Another insightful example originates from an attempt to determine the structure of the molecular ion from an all-particle computation [98–100]. The calculation of the single expectation value of the HHH angle in H is not sufficient to distinguish between the linear and triangular arrangements of the three protons, since the expectation, i.e. average value, , would be the same either for a linear or for a triangular arrangement, of the three protons.