Stationary eigenfunctions

The B-spline K-matrix method follows the close-coupling prescription a complete set of stationary eigenfunctions of the Hamiltonian in the continuum is approximated with a linear combination of partial wave channels (PWCs) [Pg.286]

The star indicates the complex conjugate. The stationary eigenfunctions Ea(Q,q) form a basis in the Hilbert space of Hmoh be., each function within this space can be uniquely represented in terms of the Ea(Q,q). 

Note that eigenfunctions with A,. n = 0 are stationary eigenfunctions and those with 0 are nonstationary. Furthermore, the stationary pi- n with n = 0 are uniform distributions on the I(p,q) = I torus, whereas the non-... 

Eigendistributions for ergodic systems may also be readily obtained, although they prove less useful than those obtained for the regular case. Specifically, stationary eigenfunctions that are L2 on the energy hypersurface for this case are given by... 

This relation emphasizes again the similarity between the properties of Floquet eigenfunctions and the ordinary stationary eigenfunctions (see Ref. [35] for more details). Although it has been proven here only for stroboscopic times, Eq. (29) may be written more generally by changing NT into an arbitrary t. 

The fundamental consequence of this assertion is that the stationary eigenfunctions of a quantum system may be found from the eigen-problems of the operators that commute with the Hamiltonian of the system. Moreover, if two operators give eigen-values on the same eigen-fimction of a quantum system,... 

If the hamiltonian operator for a system is time independent, stationary eigenfunctions exist of the form -ijriq.co) exp(—/ / ). The time-dependent exponential does... 

For the quantum mechanical results that we require we shall be concerned only with stationary states, known sometimes as eigenstates. The wave functions for these states may be referred to as eigenfunctions and the associated energies E as the eigenvalues. 

A special case of some interest is one in which the hamiltonian is stationary and the are taken to be its eigenfunctions. The matrix H is then diagonal Eq. (7-53) results in... 

For partially ordered media the stationary solution (A8.10) is an eigenfunction of the integral operator in (A8.10), belonging to the eigen-... 

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. 

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

For stationary states, the eigenfunctions xp if, Q) may be chosen to be real functions, so that this integral can also be written as... 

The stationary states of file system are described by the eigenfunctions and the eigenvalues s of the unpertuibed Hamiltonian. 

So, Eq. (3.14) with boundary conditions is the equation for eigenfunction X (x) of the nth order. For X0(x), Eq. (3.14) will be an equation for stationary probability distribution with zero eigenvalue y0 = 0, and for X (x) the equation will have the following form ... 

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

Here,, (0 for each i, are normalizable stationary state eigenfunctions of... 

Therefore, the condition that the orbitals yield a stationary point (hopefully a minimum) on the energy hypersurface with respect to variations is that the orbitals are eigenfunctions of the Fock operator, with associated orbital energy, e,... 

Note that the expectation value on the l.h.s. of Eq. (14.28) is simply 5, a, because of die orthogonality of the stationary-state eigenfunctions. Thus, only die term k = m survives, and we may rearrange the equation to... 

Ket notation is sometimes used for functions in quantum mechanics. In this notation, the function / is denoted by the symbol j/) /—1/>. Ket notation is convenient for denoting eigenfunctions by listing their eigenvalues. Thus nlm) denotes the hydrogen-atom stationary-state wave function with quantum numbers , /, and m. 

Thus the squares of the absolute values of the coefficients in the expansion (1.40) give the probabilities for observing the various possible values gf of the physical property qJfot the special case where the state function is an eigenfunction of G, F= stationary state, we have H = E 4, and an energy measurement is certain to give E. 

Exercise. Each eigenfunction Px of (2.7) produces an eigenfunction Fx of (2.11) with the same eigenvalue X, with the exception of the stationary solution P0. 

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