Photon, angular momentum

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

The electron and photon angular momentum projections, m, v, and the recoil direction, k, appearing in Eq. (A.3) are defined in the molecular frame, but our... 

This expression has the structure of quantum-mechanical expectation values, defining the operator for photon angular momentum as... 

Barth et al. performed a quantum simulation of laser-driven electron dynamics in Mg porphyrin, which is an aromatic molecule of symmetry [14]. The results of their simulation showed that n electrons of Mg porphyrin can be rotated around its aromatic ring by an ultrashort circularly polarized UV laser pulse propagating along its C4 axis. The rotation direction of n electrons is predetermined in a laboratory frame by that of the polarization plane of the circularly polarized laser pulse, that is, by photon angular momentum. 

For example, molecules that belong to low-symmetry point groups without any degenerate representations cannot have electronic or vibrational angular momentum, and therefore the photon angular momentum must be absorbed by the rotations, so A/ = 1 in that case. 

Optical tweezers have also been used to transfer various types of photon angular momentum to macroparticles, as was done long ago with the photon spin and atoms (Chapter 4). It is not out of place here to note that the photon angular momentum is, first, associated with the spin for circularly polarized light and has a magnitude of h = h/2Tt per photon. Second, there is an orbital angular momentum that is associated with the inclination of the wavefront an the laser beam (Allen et al. 1999). Unlike... 

This completes our introduction to the subject of rotational and vibrational motions of molecules (which applies equally well to ions and radicals). The information contained in this Section is used again in Section 5 where photon-induced transitions between pairs of molecular electronic, vibrational, and rotational eigenstates are examined. More advanced treatments of the subject matter of this Section can be found in the text by Wilson, Decius, and Cross, as well as in Zare s text on angular momentum. 

Atomic and molecular magnetic dipoles have to obey the angular momentum laws of quantum mechanics, since they are proportional to angular momenta. Each dipole can therefore make just a number of orientations with an applied magnetic induction B. Each allowed orientation corresponds to a different potential energy, and absorption of a photon with suitable energy may cause a change in orientation. 

Consider a threaded rod, representing a molecular enantiomer, that lies away from an observer. If the observer reaches out and spins a nut on the rod clockwise with his right hand, the nut will travel forward, away from the observer, and will shortly fly off the rod. Here, the angular momentum imparted to the nut (electron) by the observer s hand (photon) causes it to be ejected in a specific direction from the rod (molecular enantiomer) in the observer s reference frame. This is mediated by the interaction between the chiral thread of the rod and nut (the chiral molecular potential). If the rod is turned through 180° and the action repeated, the nut (electron) still departs in the same direction, away from the observer. Hence, the orientation of the rod (molecule) in the observer s frame does not alter the direction in which the nut (electron) is ejected. 

This approach was used by Elliott and co-workers to control the ionization of alkali atoms by one- and two-photon excitation. Wang and Elliott [72] measured the interference between outgoing electrons in different angular momentum states. They showed, for example, that the angular flux of the p2P and the d2D continua of Rb is determined by the phase difference... 

Monte Carlo heat flow simulation, 69-70 nonequilibrium statistical mechanics, microstate transitions, 44 46 nonequilibrium thermodynamics, 7 time-dependent mechanical work, 52-53 transition probability, 53-57 Angular momentum, one- vs. three-photon... 

To illustrate some of these principles the angular momentum of a photon will be examined [56]. Suppose a beam of circularly polarized light falls on a perfectly black absorbing surface, which not only heats up (E = hv) but also acquires a torque, on account of the angular momentum it absorbs. Circular polarization means that the probability of an elementary observation 0(P ) = The ratio of energy/torque = w(= 2m/), the angular frequency of... 

Classically, a circularly polarized light beam with angular frequency w(= 2nv) transfers angular momentum at a rate of E/w, where E is the rate of energy transfer. Considered as a beam of photons, E = Nhui/2-n, so that the angular momentum of each photon is h/2n = h. 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

To express the angular momentum M of the electromagnetic field corresponding to one photon in terms of the wave function for the photon, M is identified with the expectation value of the angular momentum in the state... 

Formula (58) shows that the angular momentum operator for the photon consists of two terms. The first term is identical with the usual quantum-mechanical operator L for the orbital angular momentum in the momentum... 

The second term s may be called the operator for spin angular momentum of the photon. However, the separation of the angular momentum of the photon into an orbital and a spin part has restricted physical meaning. Firstly, the usual definition of spin as the angular momentum of a particle at rest is inapplicable to the photon since its rest mass is zero. More importantly, it will be seen that states with definite values of orbital and spin angular momenta do not satisfy the condition of transversality. 

Since 72 = 1, the operator has two eigenvalues, 1. In summary, the photon state may be uniquely specified by giving four quantum numbers to quantify the energy u>, the angular momentum j, the component of angular momentum M and the parity A. The normalized wave function is of the form... 

It is not possible to ascribe a definite value of the orbital angular momentum to a photon state since the vector spherical harmonic YjM may be a function of different values of . This provides the evidence that, strictly speaking, it... 

If j = 0 there is only one vector spherical harmonic which is identical with the longitudinal harmonic Y 1 = nVoo- From this observation it follows that there are no transverse spherical harmonics for j = 0. It also means that the state with angular momentum zero represents a spherically symmetrical state, but a spherically symmetrical vector field can only be longitudinal. Thus, a photon cannot exist in a state of angular momemtum zero. 

The selection rule for rotational Raman transitions are AJ = 2. This result relates to the involvement of two photons, each with angular momentum h, in the scattering process. Also allowed is A J = 0, but since such a transition implies zero change in energy it represents Raleigh scattering only. 

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