Energy-momentum relation

As a direct consequence of Eq. (3.109), the four-dimensional length of the relativistic momentum has the same value in all frames of reference. 

Since the four-dimensional scalar product, with the definition of the 4-momentum in Eq. (3.110), yields 

The existence of negative-energy solutions is therefore an intrinsic feature of every relativistic theory, although they cannot be interpreted in classical mechanics and thus have to be discarded here. In quantum mechanics, we will have to find suitable means and interpretations to cope with this peculiarity (cf. chapters 5 and 7). 

The energy of a free particle as given by Eq. (3.112) may always be written as the sum of two terms. 

The second term on the right-hand side represents the familiar nonrelativistic kinetic energy and the third term is the first relativistic correction higher-order corrections have been suppressed here. We could have equally well expanded the energy-momentum relation as given by Eq. (3.118) to arrive at 

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All the ET compounds are characterized by their nearly perfect 2D band structure resulting in an almost cylindrical form of the FS. For some materials it was possible to determine quantitatively the transfer integral between the ET layers, i. e., the dispersion of the energy-momentum relation perpendicular to the highly conducting planes, the so-called warping. In some cases an... 

This gives sufficient resolution to study the geometric structure of molecules. [Since 40-keV electrons travel at a significant fraction of the speed of light, the relativistic energy-momentum relation must be used. The corrected de Broglie wavelength is actually 6.016 x 10- m.]... 

For particles with spin-1/2 we would expect (on the basis of nonrelativistic quantum mechanics) that spinors with two components would be sufficient. But the Dirac spinors have to be (at least) four-dimensional. A mathematical reason lies in the nature of the algebraic properties that have to be satisfied by the Dirac matrices a and 0 if the Dirac equation should satisfy the relativistic energy-momentum relation in the sense described above, see (6). 

The behavior of solutions of the time-dependent Dirac equation is rather strange. This can be shown explicitly for the Dirac equation in one space dimension. In this case two spinor components are sufficient, because in one space dimension the linearization E = cap + (3mc of the energy momentum relation 2 = -t- requires just one a matrix (there is only one component of... 

Here, c = l/(eo/ o) is the velocity of electromagnetic radiation in vacuum. The dependence of electromagnetic wave propagating through a crystal is called the dispersion law. Hence, Eq. (1.7) represents the dispersion law of a transverse electromagnetic wave in an infinite crystal [17]. 

It should be noted that the salient difference between the energy-momentum relation between bulk semiconductor and quantum well material is that the k vector associated with Eq takes on discrete, well-separated values. In the quantum well device, the density of states is obtained fi om the magnitude of the two-dimensional k vector associated with the y-z plane, as compared to the three-dimensional wavevector for the bulk semiconductor. As a result, the final density of states for the quantum well structure is given by... 

For ultra-relativistic particles with p me the rest energy contribution to the energy E is negligible and we therefore find E c p. The energy-momentum relation (3.118) also holds for massless particles with m = 0 and reads E = c p (for positive-energy solutions). 

The physical content of the Klein-Gordon Eq. (5.6) is only the energy-momentum relation. It does not contain a reference to the quantum mechanical spin of a particle and can therefore not explain its experimentally observed occurrence. 

The Klein-Gordon equation simply implements the relativistic energy-momentum relation for a freely moving particle. Accordingly,... 

In addition, such a power series expansion is, however, only permitted for analytic, i.e., holomorphic functions and must never be extended beyond a singular point. Since the square root occurring in the relativistic energy-momentum relation Ep of Eq. (11.11) possesses branching points at X = p/nteC = i, any series expansion of Ep around the static nonrela-tivistic limit T = 0 is only related to the exact expression for Ep for non-ultrarelativistic values of the momentum, i.e., t < 1. This is most easily seen by rewriting Ep as... 

J+ rrP-c being the relativistic energy-momentum relation. In order to keep the notation as simple as possible, we will continue to denote as being of (fc—l)-th order in V, although obviously it cannot always be... 

Near the bottom of the conduction band of a rare-gas liquid the energy momentum relation is probably parabolic and may be characterized by an effective mass. At higher energies, possibly as low as -I eV above the band minimum, a significant non-parabolic behavior is likely. This may arise, as in the solid, due to band structure effects, or alternatively due to the interaction with density fluctuations in a way reminiscent of the formation of the "bubble in liquid helium. 

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