Eigenfunctions/eigenvalues

Here,>/ is the Hamiltonian operator, which indicates that certain operations are to be carried out on a function written to its right. The wave equation states that, if the function is an eigenfunction, the result of performing the operations indicated by J( will yield the function itself multiplied by a constant that is called an eigenvalue. Eigenfunctions are conventionally denoted P, and the eigenvalue, which is the energy of the system, is denoted E. 

How rapidly an average converges will depend on the spectral properties of the operator as well as the choice of the initial distribution. For example, if we assume that Ai, pd) is some eigenvalue, eigenfunction pair with Re(Ai) < 0 and take the initial distribution to be... 

The angular momentum has three components —, and — along the X-, y-, and z-axes in a Cartesian coordinate system. These components satisfy the eigenvalue-eigenfunction equation, with eigenvalue matrix, A ... 

Substitute the corresponding eigenvalues, eigenfunctions, and random variables in Eq. 15 to obtain realizations of the random process... 

These equations in g+Cx) and g (x) are inhomogeneous and coupled to one another by the terms in tfieir right-hand sides. If we drop those terms in the second equation, we generate the eigenvalue-eigenfunction equation,... 

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

Balint-Kurti G G, Dixon R N and Marston C C 1990 The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions J. Chem. See. Faraday Trans. 86 1741... 

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... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

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 problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

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