Wave function gauge transformation

The gauge-transformed wave function may consequently be written in the manner... 

In the case of an electron wave function the required gauge field must be the electromagnetic field. To find the correct form of this compensating field we look for the transformed wave function... 

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... 

Hence, the method of Mead and Truhlar [6] yields a single-valued nuclear wave function by adding a vector potential A to the kinetic energy operator. Different values of odd (or even) I yield physically equivalent results, since they yield (< )) that are identical to within an integer number of factors of exp(/< )). By analogy with electromagnetic vector potentials, one can say that different odd (or even) I are related by a gauge transformation [6, 7]. 

The transformation T we adopt is induced by the wave function normalization condition which, in terms of the weights, reads w + W3 = 1. From (3.5), it is apparent that if T sends the vvm set into a new set wm with ivi = vvi + iv3 = 1 as one of its elements, then both the first row and the first column of the transformed polarization component of the solvent force constant matrix K, "/ = T. Kp°r. T (T = T) are zero, since the derivatives of wi are zero. Given the normalization condition and the orthogonality requirement — with the latter conserving the original gauge of the solvent coordinates framework — one can calculate T for any number of diabatic states [42], The transformation for the two state case is... 

In an environment of atoms in collision, interatomic contacts consist of interacting negative charge clouds. This environment for an atom is approximated by a uniform electrostatic held, which has a well-defined effect on the phases of wave functions for the electrons of the atom. It amounts to a complex phase (or gauge) transformation of the wave function ... 

The most general formulation of the gravitational field is in terms of gauge theory. In ordinary gauge theory a charged field is described by a complex wave function ip, whose phase changes under gauge transformation,... 

In principle, a chemical shift calculation represents a perturbation theory, because of the presence of an external field Bz and magnetic moments due to the dipole character of nuclei. Therefore, perturbations to the Hamiltonian and the wave function have to be considered. The next important point is that the origin of the vector potential Az is not fixed due to the relation Bz = rot Az- Any change of the gauge origin Rq should not change any measurable observable. Therefore, a gauge transformation of the wave function 1%) and Hamilton operator h is essential... 

Gauge transformations play an important role in both relativistic and non-relativistic QED. We have seen that E and B fields are independent of the choice of gauge although the 4-potential may have very different form in different gauges. This can have implications in approximate schemes such as Hartree-Fock, for which the wave-functions may predict radiative transition rates differing by orders of magnitude [58] when different gauges are used. 

Now we can see that the change of the gauge origin corresponds to a unitary transformation of the Hamiltonian. Although the wave function is affected by the unitary transformation, the eigenvalues remain unchanged and the Schrodinger equation is solved exactly. 

In addition, it transforms like a two-fermion wave function under rotations in position and spin space and nnder gauge transformations. The transformation properties yield a general classification scheme for the superconducting order parameter which is represented by a 2 x 2-matrix in (psendo-)spin space. It can be decomposed into an antisymmetric (s) and a symmetric (r) contribution according to 2l(k) = 2lj(k) - - f fk) with... 

If we define the vector potential of a homogeneous magnetic field B as A(r) = B X (r — Ro)/2, then the gauge origin Rq dependence can be understood as a gauge transformation with the gauge function defined by x = —(B X Ro) rl1 (compare section 2.4). This vector potential then produces terms that depend on the arbitrary position Rq- Only for exact wave functions do these terms vanish (complete basis set), while they carmot be neglected in any (small) finite one-electron basis set. 

A gauge transformation A->A = A- -VA would be expected to cause a change in the wave function, say ilr lr. We assume that this may be written as... 

Thus, the wave function must incorporate the gauge term exponentially in order for the energy to remain invariant under gauge transformations. We should add that the same factor must be included also for the nonrelativistic treatment of the magnetic fields. For a more general derivation for the nonrelativistic case, the reader should consult the book by Sakurai (1967). 

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