Schrodinger eigenvalue

We now solve the Schrodinger eigenvalue equation for the harmonic oscillator by the so-called factoring method using ladder operators. We introduce the two ladder operators d and a by the definitions... 

The Schrodinger operator can be used to make predictions about measurements of the energy of the electron in a hydrogen atom. For example, suppose (j) e satisfies the Schrodinger eigenvalue equation... 

However, we will need the Schrodinger eigenvalue equation to calculate... 

If we restrict our attention to one energy level of the Schrodinger operator, then the Schrodinger eigenvalue equation (Equation 8.16 ) holds so that... 

For a single electron, the time-independent Schrodinger eigenvalue problem is determined by the variational condition... 

Once the Hamiltonian is constructed, first principles nuclear dynamical simulations are carried out by solving the Schrodinger eigenvalue equation numerically. The spectral intensity in the photoinduced process is described by Eermi s golden rule... 

We seek to develop approximate solutions of the time-independent Schrodinger eigenvalue equation... 

While not unique, the Scluodinger picture of quantum mechanics is the most familiar to chemists principally because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section will focus on the Schrodinger fomuilation and its associated wavefiinctions, operators and eigenvalues. Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this subsection. Treatments of alternative fomuilations of quantum mechanics and discussions of relativistic effects can be found in the reading list that accompanies this chapter. 

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. 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

The eigenvalues E0, Elt E2,. .. of the Schrodinger equation (Eq. II. 1) form the electronic energies of the system under consideration. It is evident that the solution of Eq. II. 1 must involve considerable mathematical difficulties, and so far, the strongest tool we know for handling this problem is the variation principle. If the wave function W is properly normalized so that... 

If the potential energy of a system is an even function of the coordinates and if (q) is a solution of the time-independent Schrodinger equation, then the function is also a solution. When the eigenvalues of the Hamiltonian... 

The variation method gives an approximation to the ground-state energy Eq (the lowest eigenvalue of the Hamiltonian operator H) for a system whose time-independent Schrodinger equation is... 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

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