Wave equation square-integrable

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. 

Equations (22) and (23) imply a — 1 = 0. Note that the theorem is still applicable to the N-body Hamiltonian Eq. (6), and Eq. (23) is valid even when a square-integrable wave function does not exist at the threshold. 

We note that the choice of a Hilbert space of square-integrable functions as the state space of the evolution equation is perfectly natural for the Schrodinger equation. The solutions of the Schrodinger equation are in the Hilbert space L (R ) (they have only one component), and the expression tj x,t) is interpreted as a density for the position probability at time t. Hence the norm of a Schrodinger wave packet,... 

We are going to explain the procedure of forming wave packets out of plane waves for the free Dirac equation. The free stationary Dirac equation Ho tp = Erp has no square-integrable solutions at all. But it turns out that for E > me there are bounded oscillating solutions (here bounded means that the absolute value [ipix, t) of the solution remains below a certain constant M for all x and t). As for the Schrodinger equation, it is comparatively easy to find these solutions. They are similar to plane waves with a fixed momentum (wavelength), that is, they are of the form... 

This is the set of all energies for which the free stationary Dirac equation has plane-wave like solutions (out of which square-integrable wave packets can be formed). The spectrum a Ho) is the continuum of all real numbers except the numbers in the spectral gap, the open interval (—mc, mc ). 

Equation (QHQ-Eo)14 0 >= 0 (where Q is defined by Q = 4>o >< Pol)/ has fhe characfer of a self-consisfent equation, like an ordinary HF equation. This is so because QHQ depends on the wave function on which it operates, which in turn can depend implicitly on the eigenvalue Eq. Its solution is effected by successively expanding the function space spanned by the trial ho and looking for a maximal square-integrable ho still satisfying the above equation... 

In the above equation, the spin variable was not included to avoid complications at this stage. Moreover, note that a single value of the wave function P(ri,...r , 0 does not represent the state of a physical system at time t, but the whole range of such values is needed. For this reason, the state at time t will be denoted by l F(0). Further, the acceptable IT(t)) are those that belong to the space of square integrable wave functions. In the following, normalized wave functions shall be considered unless otherwise stated. Thus, ( (01 (0) = 1 or explicitly... 

A second method is try to solve the time dependent Schrodinger equation by expanding the wave function in a finite Hilbert basis set (FHBS) of square integrable functions i.e., some independent functions which go to zero for large r. The method has been developed by Bates and McCarrol [5.13],... 

This boundary condition on the Schrodinger equation may be satisfied provided the approximate wave function is expanded in a set of normalized orbitals. In practice, the orthonormal orbitals from which the determinants are constructed are expanded in a finite set of Gaussian functions (and sometimes Slater-type functions) as discussed in Chapter 6. The asymptotic form of these functions ensures that the wave function is square-integrable. 

In wave mechanics the electron density is given by the square of the wave function integrated over — 1 electron coordinates, and the wave function is determined by solving the Schrddinger equation. For a system of M nuclei and N electrons, the electronic Hamilton operator contains the following tenns. 

It should be noted that we have written E = +cVp2 + m2c2, rather than the more usual relation E2 — c2p2 + m2c4, so as to insure that the particles have positive energy. In equation (9-63), x(x,<) is a (2s + 1) component wave function whose components will be denoted by X (x,<) ( = 1,- -, 2s + 1) and the square root operator Vm2c2 — 2V2 is to be understood as an integral operator... 

Normalization is the procedure of arranging for the integral over all space of the square of the orbital wave function to be unity, as described in Section 1.3. For molecular orbitals this is expressed by the equation ... 

The function r is the bipolar square wave illustrated by waveform B, which has an amplitude of unity and is in phase with eR. We shall demonstrate that the integral of Equation 8.42 is proportional to R over the first half-cycle. As an... 

Integration over a time interval long enough compared to the period of the wave, that is, where cos2 x approaches to 1/2, gives the mean square value E(z, t)2 (Equation 6.7) ... 

From the above equation and the integration of the ak(t) values, we can calculate the expressions of the eigenfunctions. We will use the example of the square-wave potential program, since the effects of its crystalline arrangement was clearly demonstrated. First, the vibronic and electronic states of the H+-H20 system perturbed by an ac electric potential are subjected to a pulsating electric field ... 

Fig. 8.1. Response to pulsatile stimulation by extracellular cAMP in the model for cAMP synthesis based on receptor desensitization. Equations (5.16) of the two-variable model are integrated in the case where periodic stimulation by extracellular cAMP (y) takes the form of a square wave. In (a), the stimulus consists in raising y from 0 to 10 for 5 min at 5 min intervals. In (b), the same pulse is applied at 1 min intervals. In each case the variation of intracellular cAMP (j8) is represented, as well as the variation of the total fraction of active receptor (pp). Parameter values are those of fig. 5.38 (Martiel Goldbeter, 1987a).

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