The extra phase factor expresses the fact that a quantum mechanical state defines a normalised vector in Hilbert space only up to a phase factor i . e . as an element of projectivised Hilbert space.
32.
The extra phase factor expresses the fact that a quantum mechanical state defines a normalised vector in Hilbert space only up to a phase factor i . e . as an element of projectivised Hilbert space.
33.
So, for an adiabatic process, a particle starting from nth eigenstate also remains in that nth eigenstate like it does for the time-independent processes, only picking up a couple of phase factors.
34.
In the case of non-birefringent materials, however, the 2? Jones matrix is the identity matrix ( multiplied by a scalar phase factor and attenuation factor ), implying no change in polarization during propagation.
35.
In other words, there is no physical difference between two polarization states | \ psi \ rangle and e ^ { i \ alpha } | \ psi \ rangle, between which only a phase factor differs.
36.
For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schr�dinger equation.
37.
The map from paths to group elements is called the Wilson loop or the holonomy, and for a U ( 1 ) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path.
38.
Thus in consideration of this, a state | \ psi \ rangle can be encoded with this information ( i . e . the phase factor ) and thus evolve unitarily under this dephasing process, by defining the following encoded qubits:
39.
Applying this to the Lorentz group, if is a projective representation, then direct calculation using ( G4 ) shows that the induced representation on is, in fact, a proper representation, i . e . a representation without phase factors.
40.
When acting on an eigenket, the phase factor e ^ { ( \ alpha \ beta ^ *-\ alpha ^ * \ beta ) / 2 } appears in each term of the resulting state, which makes it physically irrelevant.
