Eigenfunction stationary

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. 

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

Although the above explanation relied on a crude semiclassical estimate (with exponential accuracy), it can easily be refined either by exactly solving the Schrodinger equation for the one-dimensional potential (7.1) (see, for example, Press [1981]) or, for sufficiently high barriers (V0/h(o0> 2), by employing the WKB approximation. The eigenfunctions of stationary states A and E... 

The index k(n) recalls that the nuclear fluctuation quantum states in eq.(l 1) are determined by the electronic quantum state via potential energy Een(7 )- Once the electronic problem is fully solved, via a complete set ofeq.(5), it is not difficult to see that pTif nk) multiplied by the box-normalized wave solutions (see p. 428, ref. [17] 2nd ed.) are eigenfunctions ofthehamiltonian H0and, for stationary global momentum solutions, the molecular hamiltonian is also diagonalized thereby solving eq. (2). 

The chemical reaction corresponds to a preparation-registration type of process. With the volume periodic boundary conditions for the momentum eigenfunction, the set of stationary wavefiinctions form a Hilbert space for a system of n-electrons and m-nuclei. All states can be said to exist in the sense that, given the appropriate energy E, if they can be populated, they will be. Observe that the spectra contains all states of the supermolecule besides the colliding subsets. The initial conditions define the reactants, e.g. 1R(P) >. The problem boils down to solving eq.(19) under the boundary conditions defining the characteristics of the experiment. 

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]

Here the wavefunctions i> and f> are eigenfunctions of the stationary atomic Hamiltonian, the -function ensures energy conservation, and the quantity p describes the density of final states in the photoprocess (see equ. (7.28g)). 

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). 

Alternatively, it turns out that these probabilities can be extracted from stationary scattering states, i.e., eigenfunctions of the Hamiltonian with energy E0 = pl/(2m), and the asymptotic behavior... 

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