Eigenvalue problems in quantum mechanics

If A is symmetric, so is L AL As L is lower triangular, we can compute very quickly column-by-column by forward substitution 

Once we obtain the eigenvalues Xj and eigenvectors z of L AL we compute the corresponding generalized eigenvectors 

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 

The probability of finding an electron in [x, x + dx] is ijf(x) dx, where V (x) is the wave-function of the electron, satisfying the Schrodinger equation. 

Numerical solution of a differential equation eigenvalue problem 

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As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

J. P. Modisette, P. Nordlander, J.L. Kinsey and B.R. Johnson. Wavelet Bases in Eigenvalue Problems in Quantum Mechanics, Chem. Physics Letters. 250 (1996), 485 94. 

Wavelet Bases in Eigenvalue Problems in Quantum Mechanics. 

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

Perturbation theory in general is a very useful method in quantum mechanics it allows us to find approximate solutions to problems that do not have simple analytic solutions. In stationary perturbation theory (SPT), we assume that we can view the problem at hand as a slight change from another problem, called the unperturbed case, which we can solve exactly. The basic idea is that we wish to find the eigenvalues and eigenfunctions of a hamiltonian H which can be written as two parts ... 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The characteristic-value problem - more often referred to as the eigenvalue problem - is of extreme importance in many areas of physics. Not only is it the very basis of quantum mechanics, but it is employed in many other applications. Given a Hermitian operator a, if their exists a function (or functtofts) g such that... 

While details of the solution of the quantum mechanical eigenvalue problem for specific molecules will not be explicitly considered in this book, we will introduce various conventions that are used in making quantum calculations of molecular energy levels. It is important to note that knowledge of energy levels will make it possible to calculate thermal properties of molecules using the methods of statistical mechanics (for examples, see Chapter4). Within approximation procedures to be discussed in later chapters, a similar statement applies to the rates of chemical reactions. 

Once the mathematical formalism of theoretical matrix mechanics had been established, all players who contributed to its development, continued their collaboration, under the leadership of Niels Bohr in Copenhagen, to unravel the physical implications of the mathematical theory. This endeavour gained urgent impetus when an independent solution to the mechanics of quantum systems, based on a wave model, was published soon after by Erwin Schrodinger. A real dilemma was created when Schrodinger demonstrated the equivalence of the two approaches when defined as eigenvalue problems, despite the different philosophies which guided the development of the respective theories. The treasured assumption of matrix mechanics that only experimentally measurable observables should feature as variables of the theory clearly disqualified the unobservable wave function, which appears at the heart of wave mechanics. 

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