State vector phase factor

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Heisenberg- Type Description.—Observer 0 and O ascribe to bodily the same state, the same state vector Y> (i.e., T0> = T0.> apart from possible phase factors), but describe observables by operators Q and Q, respectively, which are in one-to-one correspondence. 

It is instructive to compare the states represented by normalized vectors v and vel9 respectively. The probabilities of observing each of these vectors in the state w are uTu 2 and uTve 9 2 respectively. Whatever the choice of w, et9 2 = e10 e l6 = 1 and the two results are always the same. Any state vector therefore has associated with it an arbitrary phase factor. 

It is a bit of a lie to say, as we did in previous chapters, that complex scalar product spaces are state spaces for quantum mechanical systems. Certainly every nonzero vector in a complex scalar product space determines a quantum mechanical state however, the converse is not true. If two vectors differ only by a phase factor, or if two vectors normaUze to the same vector, then they will determine the same physical state. This is one of the fundamental assumptions of quantum mechanics. The quantum model we used in Chapters 2 through 9 ignored this subtlety. However, to understand spin we must face this issue. 

At last, after several chapters of pretending that the state space of a quantum system is linear, we can finally be honest. The state space of each quantum system is a complex projective space. The reader may wish to review Section 1.2 at this point to see that while we were truthful there, we omitted to mention that unit vectors differing by a phase factor represent identical states. (In mathematics, as in life, truthful and honest are not synonyms.) In the next section, we apply our new insight to the spin state space of a spin-1/2 particle. 

Another representation of the basis Ii> is I 0> 10k> where I0k> is an orthogonal basis for the orthogonal complement to 10>. Neglecting the phase factor an arbitrary normalized state vector becomes... 

This result is an exact expression for the transition matrix element. Physically we have a dipole interaction with the vector potential and a dipole interaction with the magnetic field modulated by a phase factor. The problem is that this integral is difficult to compute. An approximation can be invoked. The wavevector has a magnitude equal to 1 /X. The position r is set to the position of an atom and is on the order of the radius of that atom. Thus K r a/X. So if the wavelength of the radiation is much larger than the radius of the atom, which is the case with optical radiation, we may then invoke the approximation e k r 1 + ik r. This is commonly known as the Bom approximation. This first-order term under this approximation is also seen to vanish in the first two terms as it multiplies the term p e. A further simplification occurs, since the term a (k x e) has only diagonal entries, and our transition matrix evaluates these over orthogonal states. Hence, the last term vanishes. We are then left with the simplified variant of the transition matrix ... 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

LDA methods have been employed to investigate solids in two types of approaches. If the solid has translational symmetry, as in a pure crystal, Bloch s theorem applies, which states that the one-electron wave function n at point (r + ft), where ft is a Bravais lattice vector, is equal to the wave function at point r times a phase factor ... 

In other words, the Hamiltonians Em just introduce a global phase factor in front of the initial information state vector, but leaves its coherence intact. Obviously, the correction conditions (15b) are obtained as a particular case of the above conditions, setting Cm = 0 for all m. Yet, though less general, they will be employed in the rest of the paper for the sake of simplicity. 

The development of electronic geometric phase factors is governed by an adiabatic vector potential induced in the nuclear kinetic energy when we extend the Born-Oppenheimer separation to the degenerate pair of states [1, 23-25]. To see how the induced vector potential appears, we consider the family of transformations which diagonalize the excited state electronic coupling in the form... 

The operator within the parentheses remains the same if (r - - na) is substituted for p. The Bloch theorem (231, 437) states that because of this periodicity, iy(r -f a) must be the same as to within a phase factor, that is (r - - a) = (r). Each eigenfunction that satisfies Eq. (2) has a wavevector associated with it such that translation by a lattice vector na is equivalent to multiplying the eigenfunction, v (r), by the phase factor exp(/ifc a). 

The periodicity of the nuclei in the system means that the square of the wave function must display the same periodicity. This is inherent in the Bloch theorem (eq. (3.75)), which states that the wave function value at equivalent positions in different cells are related by a complex phase factor involving the lattice vector t and a vector in the reciprocal space. 

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