By Carroll B.W., Ostlie D.A.
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Additional resources for An introduction to modern astrophysics
10) Upon traversing the grids, a constant energy eV needs to be added to the mass-energy m(dt/'dr) to maintain the canonical energy Jco a constant. 12) This ratio being independent of the incidence angle, sin(/)/sin(/') is a constant, as is the case for nonrelativistic particles. Two essential pos tulates were made in the above discussion: medium isotropy and invariance of the canonical momentum in directions of translational invariance of the medium. In a wave theory, the latter follows from the proportionality of the canonical momentum to the wave vector.
This analogy is exemplified in Fig. 1-27, where we compare the trajectory of a ball in a shallow gutter and the trajectory of a light ray in a two-dimensional graded-index fiber. If V (respectively, k2 or n2) is quadratic in x, for example, the rays are (exactly) sinusoids in both systems. 8b), which has the dimension of length, does not, in general, represent the time of flight of optical pulses. What we are comparing here are trajectories in space. Time-dependent concepts have been used in the discussion because the dispersive properties of massive particles happen to be known from Newton's dynamics.
These argu ments are not completely satisfactory because appeal must be made to the second quantization (E = hco). 46 For scalar waves, the proof follows from the expression 44 1 Description of Optical Beams ( 3 £ / 3 K ) K of the canonical stress-energy density, where £ denotes the average Lagrangian density. 47 The quantity denoted J is the flux of 3 £ / 3 k , the spatial component of the wave action, through the cross section of the beam. Thus J is adiabatically invariant in lossless media as stated earlier.
An introduction to modern astrophysics by Carroll B.W., Ostlie D.A.