Integrability, quadratic

In order for (jc, i) to satisfy equation (2.9), the wave funetion must be square-integrable (also ealled quadratically integrable). Therefore, W(x, /) must go to zero faster than 1 / Z x[ as x approaches ( ) infinity. Likewise, the derivative dW/dx must also go to zero as x approaehes ( ) infinity. 

To be a suitable wave function, Sxiip) must be well-behaved, i.e., it must be continuous, single-valued, and quadratically integrable. Thus, pSu vanishes when p oo because Sxi must vanish sufficiently fast. Since Sxi is finite everywhere, pSxi also vanishes at p = 0. Substitution of equations (6.22) and (6.23) into (6.19) shows that Sxiip) is normalized with a weighting fianction w(p) equal to p ... 

Recall that a function f(x) is a rule that associates a number with each value of the variable x. A functional F[f] is a rule that associates a number with each function /. For example, the functional F[/] = f (x)f(x)dx associates a number, found by integration of / 2 over all space, with each quadratically integrable function f(x). The variational integral... 

These postulates place certain constraints on what constitutes an acceptable wave function. For a bound particle, the normalized integral of q> - over all space must be unity (i.e., the probability of finding it somewhere is one) which requires that F be quadratically integrable. In addition, q must be continuous and single-valued. 

Carrying out a power-series solution, one finds that the quadratically integrable normalized solutions of (1.132) are... 

These manipulations have brought us to a familiar equation we recognize (4.23) as the Schrodinger equation (1.132) for a one-dimensional harmonic oscillator with force constant ke. Before we can conclude that (4.23) and (1.132) have the same solutions, we must verify that the boundary conditions are the same. For quadratic integrability, we require that S(q) vanish for q = oo. Also, since the radial factor F(R) in the nuclear wave function is... 

If / 7(r) is a vector of functions, one for each open channel, that are regular at the coordinate origin and quadratically integrable, then... 

The X multipliers are determined by the orthogonality condition. If the quadratically integrable radial basis in channel p is the orthonormal set r]pa, values of the Lagrange multipliers are given by... 

This is transfer covariant if all quadratically integrable functions are represented in the same orbital basis. Requiring fps to be orthogonal to all radial factor riPa(r)) enforces a unique representation, but introduces Lagrange multipliers in the close-coupling equations. An alternative is to require... 

One should also be aware of the fact that for the general complex transformation V, there may exist wave functions VF = F(A ) defined on the real axis which are quadratically integrable, but which cannot be analytically continued in such a way that the symbol L/VP becomes meaningful. In discussing an unbounded operator U and its domain D(U), it may then be practical to introduce the complement C(U) only with respect to the part of the L2 Hilbert space, for which the Operator U may be properly defined. A more detailed discussion of these problems is outside the scope of this review. In the following discussion, we will not further specify the form of the transformation U. 

According to the first postulate, the state of a physical system is completely described by a state function fifiq, /) or ket T1), which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q, q2, , so that the state function may also be written as q, q2, , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector XV is a function of x and t Tfix, /). For a particle or system in three dimensions, the components of q are x, y, z and I1 is a function of the position vector r and t Tfi r, /). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable. 

Postulate 1. The state of a system is described by a function i of the coordinates and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. i j is single-valued, continuous, and quadratically integrable. 

A regular function is a mathematical function satisfying the three conditions of being (i) single-valued (ii) continuous with its first derivatives and (iii) quadratically integrable, i.e. vanishing at infinity. 

Simpson s 1/3 rule is based on quadratic polynomial interpolation. For a quadratic integrated over two Ax intervals that are of uniform width or panels, we can express the area as ... 

Since this type of equation cannot be solved by quadratic integrations, the following substitution can... 

The requirement expressed by Eqs. (20.4) and (20.6), namely that the wave function be quadratically integrable, imposes severe restrictions on ij/. The wave function must be single-valued, continuous, and may not have singularities anywhere of a character that result in the nonconvergence of the integral in Eq. (20.6). In particular, at the extremes of the Cartesian coordinates, x = + co, y = + co, and z = + oo, the wave function, as well as well as its first derivative, must vanish. 

Case I. The parameter n is either nonintegral or is a negative integer. If n is nonintegral, the solution of Eq. (21-41) behaves as for large values of and we would have ij/ = for large values of This function is not quadratically integrable... 

The grand conclusion from all of this is that the condition of quadratic integrability requires n to be a positive integer or zero. Looking back, we realize that n governs the energy through Eq. (21.38), which can be written... 

The only solutions, L(x), of this equation that are quadratically integrable are those for which the coefficient of L is zero or a positive integer this condition requires that the parameter n be an integer such that n — (I 1) > 0 orn > I 1. Since the least value of / is zero we have the quantization conditions... 

QUADRATIC INTEGRALS IN INVERSE PROBLEMS WITH MULTIPLE SCATTERING... 

Rybicki s quadratic integrals were generalized by Ivanov [6] in such a way that they relate the radiation fields at two different optical depths in an atmosphere. Ivanov showed that by using these integrals for determining... 

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