Wave function conditions

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... 

The total molecular system wave function is subject to the boundaty conditions... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles... 

The values produced may be random and not bounded within any upper or lower limits. This may happen if the boundary conditions on the total wave function are violated. 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

In almost all cases X is unaffected by any changes in the physical and chemical conditions of the radionucHde. However, there are special conditions that can influence X. An example is the decay of Be that occurs by the capture of an atomic electron by the nucleus. Chemical compounds are formed by interactions between the outer electrons of the atoms in the compound, and different compounds have different electron wave functions for these outer electrons. Because Be has only four electrons, the wave functions of the electrons involved in the electron-capture process are influenced by the chemical bonding. The change in the Be decay constant for different compounds has been measured, and the maximum observed change is about 0.2%. 

The electron wave function becomes localised in the top and bottom part of the cylindrical surface where the effective magnetic field perpendicular to the tube surface is the largest. Thus the boundary condition along the circumference direction becomes less important in high magnetic fields as has been discussed in Sec. 2. Consequently the distinction between metallic and semiconducting... 

The bra n denotes a complex conjugate wave function with quantum number n standing to the of the operator, while the ket m), denotes a wave function with quantum number m standing to the right of the operator, and the combined bracket denotes that the whole expression should be integrated over all coordinates. Such a bracket is often referred to as a matrix element. The orthonormality condition eq. (3.5) can then be written as. 

For a variationally optimized wave function, the first term is again zero (eq. (10.24)). Furthermore, the second term, which involves calculation of the second derivative of the wave function with respect to the parameters, can be avoided. This can be seen by differentiating the stationary condition (10.24) with respect to the perturbation. 

The idea is to construct a Lagrange function which has the same energy as the non-variational wave function, but which is variational in all parameters. Consider for example a CL wave function, which is variational in the state coefficients (a) but not in the MO coefficients (c) (note that we employ lower case c for the MO coefficients, but capital C to denote all wave function parameters, i.e. C contains both a and c), since they are determined by the stationary condition for the HF wave function. 

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

In Section II.D(4c), it was pointed out that, in treating correlation effects in a molecular system, it is of essential importance that the improved wave function leads to an energy curve having correct asymptotic behavior for separated atoms. It has been shown (Frost, Braunstein, and Schwemer 1948) that this condition may be fulfilled by a convenient choice of a correlation factor g. Let us consider the H2 molecule and a wave function of the type... 

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

However, an acceptable wave function must satisfy the subsidiary condition (9-475), i.e., p- = 0. 

We shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

It therefore follows from the transversality condition [Eq. (9-516)] that a physically admissible wave function u(k) can only be spacelike or a multiple of k , since k is a null vector. In the coordinate system in which... 

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