Eigenvalue analysis quantum mechanics

On analysis by Bell [30] the proof was shown to rely on the assumption that dispersion-free states have additive eigenvalues in the same way as quantum-mechanical eigenstates. Using the example of Stern-Gerlach measurements of spin states, the assumption is readily falsified. It is shown instead that the important effect, peculiar to quantum systems, is that eigenvalues of conjugate variables cannot be measured simultaneously and therefore are not additive. The uniqueness proof of the orthodox interpretation therefore falls away. 

The continuous spectrum is also present, both in physical processes and in the quantum mechanical formalism, when an atomic (molecular) state is made to interact with an external electromagnetic field of appropriate frequency and strength. In conjunction with energy shifts, the normal processes involve ionization, or electron detachment, or molecular dissociation by absorption of one or more photons, or electron tunneling. Treated as stationary systems with time-independent atom - - field Hamiltonians, these problems are equivalent to the CESE scheme of a decaying state with a complex eigenvalue. For the treatment of the related MEPs, the implementation of the CESE approach has led to the state-specific, nonperturbative many-electron, many-photon (MEMP) theory [179-190] which was presented in Section 11. Its various applications include the ab initio calculation of properties from the interaction with electric and magnetic fields, of multiphoton above threshold ionization and detachment, of analysis of path interference in the ionization by di- and tri-chromatic ac-fields, of cross-sections for double electron photoionization and photodetachment, etc. 

We shall encounter numerous situations in which eigenvalue analysis provides insight into the behavior and performance of an algorithm, or is itself of direct use, as when estimating the vibrational frequencies of a structure or when calculating the states of a system in quantum mechanics. The related method of singular value decomposition (SVD), an extension of eigenvalue analysis to nonsquare matrices, is also discussed. 

Eigenvalue analysis lies at the heart of quantum mechanics. Here we consider only a simple example involving a single electron in one dimension, but the numerical approach is the same as that used in more reahstic 3-D calculations of atoms and molecules. We wish to... 

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