Four-vectors momentum

We know that the four-vector momentum and the four-vector potential are conserved quantities under a Lorentz transformation, so we expect that a is also conserved, as may be shown with the conditions below. 

The defining equations (9-133) and (9-134) for H and P reflect the fact that the particles are free and do not interact with one another. The total energy and total momentum of the system is, therefore, the sum of energies and momenta of the individual particles as indicated by Eqs. (9-135) to (9-137). We shall see that we may consider the operators H, P as the time and space components of the four-vector P ... 

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... 

The last feature requires a new definition and formulation of SSP or FM in relativistic systems since spin is no more a good quantum number in relativistic theories spin couples with momentum and its direction changes during the motion. It is well known that the Pauli-Lubanski vector W1 is the four vector to represent the spin degree of freedom in a covariant form,... 

As a simple introduction to the subject [27], let us consider the four angular momentum vectors illustrated in figure 1.9. They are as follows ... 

In the theory of relativity, space and time variables can be combined into a four-vector with xi = x,X2 = y, x = z, JC4 = ict. The momentum and energy analogously combine to a 4-vector with p = Px, P2 = Py, P3 = Pz> PA = iE/c.By suitable generalization of the quantization prescription for momentum components, deduce the time-dependent Schrodinger equation ... 

The momentum four-vector as measured by a stationary observer, for a particle moving with relative velocity v is... 

The energy and momentum of a particle in relativistic mechanics can be represented as components of a four-vector with... 

Note the odd feature, that although q pointed along OZ, g increases as S°° moves faster along OZ. The reason is that q is not the four-momentum of a particle. Indeed it is a space-like four-vector. 

From our experience this far with vector lengths and velocities, we do not expect the magnitude of ordinary linear momentum to be invariant under the Lorentz transformations. By analogy with our previous derivation of the four-vector, we can take a cue from the relations for light signals. For photons we know that the relation... 

In terms of the momentum four-vector introduced earlier, this yields... 

However, if we are concerned about Lorentz invariance, we should at this point remember that the scalar potential is only one component of a four-vector A = (A, i/c). If the scalar potential modifies %, or equivalently E, then we would expect the vector potential to modify the momentum, which accounts for the remaining components of the four-vector. 

As we have seen, the vector properties of molecular collisions offer much richer information than that provided by scalar properties, such as the total cross-section of a reaction or the energy content of the reaction products. To illustrate this point, consider a simple atom-transfer reaction, which will be abstractly written as A -f BC AB -I- C. For this process, we can readily identify four vectors. These are the initial relative velocity v of the reagents (A, BC), the final relative velocity v of the products (AB, C), the initial rotational angular momentum of the reagent molecule BC, denoted by j, and the final rotational angular momentum of the product molecule AB, denoted by j. Here we have assumed, for simplicity, that no photons are emitted or absorbed in the collision process, and that electronic or nuclear spin angular momenta are non-existent or are randomly oriented and do not couple to other angular momenta present. A simple example of such a case would be the atom-transfer reaction O -F CS CO + S. 

Apart from a factor c, the norm of the energy-momentum four-vector. 

This idea is motivated by the fact that energy and momentum are the components of a four-vector, a relativistic quantity that transforms like a vector when subjected to a Lorentz transformation ... 

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... 

We want to divide the components of the momentum vector by po and think of the result as coordinates on a hyperplane, which we project stereographi-cally onto the unit sphere in four-dimensional Euclidean space. The Cartesian coordinates on the sphere are... 

The equation for the value of the velocity at each node is based on a momentum balance for each control volume. In the interior of the domain, the control volume has a momentum flux crossing each of the four sides. The flux depends on the sign of the velocity gradient and the outward-normal unit vector that defines the face orientation. In discrete, integral form, the two-dimensional difference equation emerges as... 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

Circular orbits are defined by n = 0. The principal quantum number specifies energy shells. For n = 1 the only solution is n = 0, k = 1, which specifies two orbits with angular momentum vectors in opposite directions. The solutions n = 0, k = 2 and n = 1, k = 1 define 8 possible orbits, 4 circular and 4 elliptic. The angular momentum vectors of each set are directed in four tetrahedral directions to define zero angular momentum when fully occupied. Taken together, these tetrahedra define a cubic arrangement, closely related to the Lewis model for the Ne atom. 

In the first place, we shall find that the four quantities Ty px0y py0y pz0 must be constant at all points of space, for equilibrium. By comparison with Eq. (2.4) of Chap. IV, the formula for the Maxwell distribution of velocities, we see that T must be identified with the temperature, which must not vary from point to point in thermal equilibrium. The quantities pxo, pyo, p 0 are the components of a vector representing the mean momentum of all the molecules. If they are zero, the distribution (2.15) agrees exactly with Eq. (2.4) of Chap. IV. If they are not zero, however, Eq. (2.15) represents the distribution of velocities in a gas with a certain velocity of mass motion, of components pxo/my pyQ/my pzo/m. The quantities px — pxo, etc., represent components of momentum relative to this momentum of mass motion, and the relative distribution of velocities is as... 

Momentum distributions of protons are derived from ECS spectra. Fig. 13 shows the four measured distributions J(y) obtained from ECS with electrons of 15, 20, 25 and 30 keV, together with NCS data. These energies correspond to momentum transfers of 47.6, 55.1, 61.8 and 67.8 A-1, respectively. The agreement, within experimental error, confirms that both experiments reveal the same physical quantity the (projection on the scattering vector q of the) momentum density distribution of protons. The NCS and ECS results of Fig. 13 can be compared directly as the experimental energy resolution contributes negligibly to the width of the spectra obtained by other technique. 

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