Photons momentum

It can be shown that when the normalization condition for the photon wave function (50) is satisfied, the vector p can be interpreted as the expectation value of the photon momentum. To do this it is necessary to express p in terms of fk. Substitution of (37) into (51) yields... 

Photon momentum., 14 834 Photon recycling, 14 846 Photon scanning tunneling microscopy (PSTM), 16 503... 

Figure 1 Angles 6 and

In this equation, the spherical angles 6 and

defined relative to the photon momentum k, photoelectron momentum p, and photon polarization vector e, as indicated in Figure 1, fi i is a dipole photoelectron angular distribution parameter, yni and Sni are nondipole photoelectron angular distribution parameters. 

As photon momentum p = E/c, the quantum assumption E = hu implies that p = hu/c = h/X. This relationship between mechanical momentum and wavelength is an example of electromagnetic wave-particle duality. It reduces the Compton equation into ... 

It is well known that the electronic structure of molecules or solids can be directly connected to optical properties, our physical observables, through the complex dielectric constant at optical frequencies. A detailed analysis of this subject is discussed in several textbooks [18]. Here we shall only highlight the main results. From the solid state point of view, direct optical transitions, i.e., transitions in which the energy tko is conserved and the photon momentum is negligible, are related to the imaginary part of the complex dielectric constant = i + i 2 by... 

The annihilation characteristics of a positron in a medium is dependent on the overlap of the positron wavefunction with the electron wavefunction [9]. From a measurement of the two photon momentum distribution, information on the electron momentum distribution can be obtained and this forms the basis of extensive studies on electron momentum distribution and Fermi surface of solids [9]. In the presence of defects, in particular, vacancy type defects, positrons are trapped at defects and the resultant annihilation characteristics can be used to characterize the defects [9, 10], Given these inherent strengths of the technique, in the years following the discovery HTSC, a large number of positron annihilation experiments have been carried out [11, 12]. These studies can be broadly classified into three categories (1) Studies on the temperature dependence of annihilation characteristics across Tc, (2) Studies on structure and defect properties and (3) Investigation of the Fermi surface. In this chapter we present an account of these investigations, with focus mainly on the Y 1 2 3 system (for an exhaustive review, see Ref. 11). 

The photon s rest mass is zero. However a photon of energy E travelling at the speed c can be considered to have acquired an ejfective mass m given by Eq (2.11). Equating Eqs (2.9) and (2.11) for the photon energy and taking the photon momentum to be p = me, we obtain... 

Because the photon momentum is negligible compared to the electron crystal momentum, the momentum conservation requirement simplifies to ... 

The delta function in Eq. (59) serves to ensure energy conservation, while the factor (r M — M) signifies the extent to which photon momentum is conserved, as determined by wavevector matching. As shown previously [1,38], we have r M = M2 in the limit of exact wavevector matching. Now, substituting Eqs. (55) and (58) into (59) allows the rate of coherent harmonic production, into an infinitesimal solid angle dfl, to be expressed as follows ... 

It is helpful to compare the average carrier momentum with the photon momentum. A carrier with kinetic energy kT and effective mass m has the momentum... 

Calculation of the carrier momentum (using Eq. (1.5)) and the photon momentum (using Eq. (1.6)) yields that the carrier momentum is orders of magnitude lar-... 

Performing the angular integration over directions of the photon momentum, summation over polarizations, and subtracting from Eq (181) the corresponding expression for the free electron (mass renormalization) one obtains ... 

While the one-loop diagram of Fig. 2c is also associated with the structure problem, here we take it together with Fig. 2d, which is not, and evaluate them as a unit. The loop is associated with an integration over the fourth component of photon momentum z, and it is straightforward to derive for Fig. 2c, which we call the ladder (L)... 

Describe the wave-particle duality of matter and energy and the theories and experiments that led to it (particle wavelength, electron diffraction, photon momentum, uncertainty principle) ( 7.3) (SP 7.3) (EPs 7.27-7.34)... 

Optohydrodynamics Fluid Actuation by Light, Fig.1 (a) Variation of photon momentum at the interface between two dielectric liquids (i) downward propagating photon and (ii) upward propagating photon. Plain and dotted arrows, respectively, represent photon momentum and the momentum transferred by the photrai to the interface. (b) Picture of the defoimatirai of a near-taitical interface [3] used to illustrate the beam propagation in cases (i) and (ii) with an illustration of the notations... 

The temperature given by = Ay/2 is not the minimum temperature achievable with laser cooling. One can begin cooling on a broad transition before switching to a more forbidden transition. Thus, atoms can be initially cooled quickly when it is important to scatter many photons/sec. Once cooled to less than 1 mk, a more leisurely cooling process can be used to reach temperatures below 10 K. At these temperatures, the recoil due to a single photon momentum becomes a factor. 

Figure 11.7 Index-matching geometry for conservation of photon momentum in CARS, 2k, = ka + kg. The experimental angle between the laser beams at frequencies a/, and uj must be adjusted to the value 9 given in Eq. 11.30 for observatipn of CARS the scattered anti-Stokes signal emerges in the well-defined direction...

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