Eigenvalues and eigenfunctions

In general, the function cp obtained by the application of the operator A on an arbitrary function ip, as expressed in equation (3.1), is linearly independent of Ip. However, for some particular function 0i, it is possible that 

A simple example of an eigenvalue equation involves the operator mentioned in Section 3.1. When operates on e, the result is 

the exponentials e are eigenfunctions of Dx with corresponding eigenvalues k. Since both the real part and the imaginary part of k can have any values from —00 to +00, there are an infinite number of eigenfunctions and these eigenfunctions form a continuum of functions. 

Another example is the operator Z) acting on either sin nx or cos nx, where is a positive integer (n 1), for which we obtain 

The functions sin nx and cos nx are eigenfunctions of with eigenvalues — n. Although there are an infinite number of eigenfunctions in this example, these eigenfunctions form a discrete, rather than a continuous, set. 

Let us repeat this calculation, using only operator equations  

Suppose that the effect of operating on some function f x) with the operator A is simply to multiply/(x) by a certain constant k. We then say that/(x) is an eigenfunction of A with eigenvalue k. As part of the definition, we shall require that the eigenfunction f x) is not identically zero. By this we mean that, although j x) may vanish at various points, it is not everywhere zero. We have 

EXAMPLE if f x) is an eigenfunction of the linear operator A and c is any constant, prove that cf x) is an eigenfunction of A with the same eigenvalue asf x). 

Write down the given information and translate this information from words into equations. 

Write down what is to be proved in the form of an equation or equations. 

The Schrodinger equation, when written as lEj/ = Exp, is an example of a special class of equations which can be put into the general form  

A is an operator which operates on the function 0 to give the same function back again, multiplied by a constant a . Any function which satisfies the equation is known as an eigenfunction of the operator A ( eigen is the German word for characteristic ), and the constant a is said to be an eigenvalue of the operator A. 

In general, there will be many functions 0 which satisfy the equation, each with its own eigenvalue. If we indicate each separate solution by the index n , the equation can be written as  

However, sin2x is notan eigenfunction of d/dx, since d/dx) sm2x) = 2cos2x, which is not a constant times sin 2x. 

(a) Manipulate the given equations of step 1 so as to transform than to the desired equations of step 2. (b) Alternatively, start with one side of the equation that we want to prove and use the given equations of step 1 to manipulate this side until it is transformed into the other side of the equation to be proved. 

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The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

The Eigenfunctions and Eigenvalues for Special Cases a. Spherical Tops... 

The exact eigenfunctions and eigenvalues can now be expanded in a Taylor series in X. 

The potential V r) in (2) being identical for all atoms in the extended cell, the problem of finding the energy eigenfunctions and eigenvalues in (8) may be reduced to their calculation for a one-atom cell. 

The eigenfunctions and eigenvalues of Eq. (10-405) can be obtained as in the nonrelativistic case upon introducing the variables... 

The search for eigenfunctions and eigenvalues in the example of the simplest difference problem. The method of separation of variables being involved in the apparatus of mathematical physics applies equelly well to difference problems. Employing this method enables one to split up an original problem with several independent variables into a series of more simpler problems with a smaller number of variables. As a rule, in this situation eigenvalue problems with respect to separate coordinates do arise. Difference problems can be solved in a quite similar manner. 

Before giving further motivations, we would like to recall the basic aspects concerned with the elementary problem of determining eigenfunctions and eigenvalues for the differential equation... 

Thus, we have obtained the eigenfunctions and eigenvalues of problem (14). A brief survey of their properties is presented below. 

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

The first step in the solution of equation (10.28b) is to hold the two nuclei fixed in space, so that the operator drops out. Equation (10.28b) then takes the form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunctions and eigenvalues of He in equation (10.6) depend only on the fixed value R of the intemuclear distance, so that we may write them as tpKiy, K) and Sk(R). If equation (10.6) is solved repeatedly to obtain the ground-state energy eo(K) for many values of the parameter R, then a curve of the general form... 

Chapters 4, 5, and 6 discuss basic applications of importance to chemists. In all cases the eigenfunctions and eigenvalues are obtained by means of raising and lowering operators. There are several advantages to using this ladder operator technique over the older procedure of solving a second-order differ-... 

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

Calculating Eigenfunctions and Eigenvalues of the Schroedinger Equation on a Grid. 

Calculate Eigenfunctions and Eigenvalues in a Given Energy Range. 

The energies before and after the interaction are shown in Fig. 5. The eigenfunctions and eigenvalues are listed in Table 1. 

Table 1. Eigenfunctions and eigenvalues of the interaction problem involving do and the four determinants which can be formed from an i - T orbital promotion...

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