We shall demonstrate that the integral of Equation 8.42 is proportional to R over the first half-cycle. The function r is the bipolar square wave illustrated by waveform B, which has an amplitude of unity and is in phase with eR. For molecular orbitals this is expressed by the equation. 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. 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. 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. For a system of M nuclei and N electrons, the electronic Hamilton operator contains the following tenns. 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. The asymptotic form of these functions ensures that the wave function is square-integrable. 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. This boundary condition on the Schrodinger equation may be satisfied provided the approximate wave function is expanded in a set of normalized orbitals. The method has been developed by Bates and McCarrol. Ī 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. In the following, normalized wave functions shall be considered unless otherwise stated. Further, the acceptable IT(t)) are those that belong to the space of square integrable wave functions. For this reason, the state at time t will be denoted by l F(0). 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. In the above equation, the spin variable was not included to avoid complications at this stage.
Bipolar square wave equation trial#
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. This is so because QHQ depends on the wave function on which it operates, which in turn can depend implicitly on the eigenvalue Eq. Įquation (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. The spectrum a Ho) is the continuum of all real numbers except the numbers in the spectral gap, the open interval (-mc, mc ).
Bipolar square wave equation free#
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). But it turns out that for E > me there are bounded oscillating solutions (here bounded means that the absolute value The free stationary Dirac equation Ho tp = Erp has no square-integrable solutions at all. We are going to explain the procedure of forming wave packets out of plane waves for the free Dirac equation. Hence the norm of a Schrodinger wave packet. 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. 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. (23) is valid even when a square-integrable wave function does not exist at the threshold. Note that the theorem is still applicable to the N-body Hamiltonian Eq. Therefore, W(x, /) must go to zero faster than 1 / Z xĮquations (22) and (23) imply a - 1 = 0. In order for (jc, i) to satisfy equation (2.9), the wave funetion must be square-integrable (also ealled quadratically integrable).
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