Eigenvalue, eigenfunction

Understand the basis of the eigenvalue problem and identify eigenfunctions, eigenvalues and operators... 

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. 

The matrix elements ot — Ej and p are not variables in the minimization procedure they are constants of the secular equations with units of energy. Note that all elements in the matrix and vector are real numbers. The vector is the set of coefficients for one eigenfunction corresponding to one eigenvalue, Ej. From Eq. (7-24),... 

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

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