Energy-momentum vector

The energy-momentum vector is the value p = T(v) M for v of the energy-momentum (Tetrode) tensor pT. We have shown (see [9]) that it is in the form... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6iV dimensions three components of p and the three components of q for each atom. If we use the symbol F to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of r and is often written as A(F). As the system evolves in time then F will change and so will A(F). 

We have noted that if is the energy-momentum four vector of a photon (i.e., P = 0, k0 > 0) there exist only two other linearly independent vectors orthogonal to ku. We shall denote these as tft k) and ejf fc). They satisfy... 

Consider next a photon of definite energy-momentum ku. Let its state of polarization be denoted by ( ). This vector can be decomposed along efi k) and e (k)... 

Since the operators P commute with one another we can choose a representation in which every basis vector is an eigenfunction of all the P s with eigenvalue It should be noted that the specification of the energy and momentum of a state vector does not uniquely characterize the state. The energy-momentum operators are merely four operators of a complete set of commuting observables. We shall denote by afi the other eigenvalues necessary to specify the state. Thus... 

In addition to CO(v = 0—2,7) populations, Houston and Kable recorded CO Doppler profiles to measure the translational energy release, and the vector correlation between the recoil velocity vector and the angular momentum vector of CO. Together, these data paint a compelling picture that two pathways to CH4 + CO are operative. The rotationally hot CO population (85% of total CO)... 

To summarize, suppose that all possible states for any given observable (spin, polarization, energy, momentum, etc.) are known and that each can be formulated in terms of a column vector a = a,i, <22, an. These vectors form an orthonormal set and are represented by an n x n matrix... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

The Fock transformation of variables consists in projecting the momentum vector p with coordinates Px, py, pz and modulus p in momentum space on a tetradimensional hypersphere of unit radius. The momentum pq = V-2E is directly related to the energy spectrum. A point on the hypersphere surface has coordinates ... 

XII. Finally, the energy momentum of the field on the 0(3) level is a 12-vector ... 

These various considerations point toward the 0(3) definition of the energy-momentum 4-vector ... 

There are therefore three energy-momentum 4-vectors present ... 

AUTOIONIZATION (or Preionization). Some bound states of atoms have energies greater than the ionization energy. An atom that is in a disciete energy state, above the. ionization point can ionize itself automatically with no change in its angular momentum vectors if there is a continuum with exactly the same characteristics. This process is called autoionization. 

Fig. 3. Magnitudes and directions of the angular momentum vectors for an / = I. s = 5 electron in an atomic energy state...

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