Momentum generalised

The quantity p — QA is called a generalised momentum. It appears in both classical electromagnetism and quantum mechanics. In the Schrbdinger picture, we make the substitution... 

Canonically-conjugate observables do not commute. Corresponding to a generalised position coordinate q there is a generalised momentum p. The commutation law is... 

P,P pressure (gas or osmotic) spectrum symbol generalised momentum (aggregate normal force, total pressure) (intensity of normal stress) polarisation (electric). 

Energy domain Effort e Row / Generalised momentum p Generalised displacement q... 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

This important equation is known as the Klein-Gordon equation, and was proposed by various authors [6, 7, 8, 9] at much the same time. It is, however, an inconvenient equation to use, primarily because it involves a second-order differential operator with respect to time. Dirac therefore sought an equation linear in the momentum operator, whose solutions were also solutions of the Klein-Gordon equation. Dirac also required an equation which could more easily be generalised to take account of electromagnetic fields. The wave equation proposed by Dirac was [10]... 

The discussion of angular momentum conservation is based on the generalised angular momentum tensor (compare [27, 28]),... 

Here we have chosen to define the generalised angular momentum tensor via the canonical energy-momentum tensor 0" ,... 

The tensor product is the tensor generalisation of the basic angular momentum coupling definition (3.84,3.85). We combine two tensors r , and Sq to form a tensor Tq in the following way. 

In the previous part of the chapter we expressed the problem of an electron in a local, central potential in terms of radial equations and eigenstates of orbital angular momentum. In generalising to the case where the electron obeys the Dirac equation (3.154) we remember that spin and orbital angular momentum are coupled. 

The above description of the excited states in terms of excitation amplitudes is frame and basis set dependent. A more convenient description is in terms of state multipoles. It can be generalised to excited states of different orbital angular momentum and provides more physical insight into the dynamics of the excitation process and the subsequent nature of the excited ensemble. The angular distribution and polarisation of the emitted photons are closely related to the multipole parameters (Blum, 1981). The representation in terms of state multipoles exploits the inherent symmetry of the excited state, leads to simple transformations under coordinate rotations, and allows for easy separation of the dynamical and geometric factors associated with the radiation decay. 

The standard random walk problem in physics is the Ornstein-Uhlenbeck process, which is a model of the Brownian motion in a dissipative medium. We are now looking at the possibility to generalise this to the quantum mechanical dynamics. To this end we introduce the one-dimensional canonical variables [x, p = ih, where we retain the quantum constant for dimensional reasons. We assume that these co-ordinates are physical in the sense that the laboratory positions are given by x and the physical forces are supposed to act on the momentum p only. 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

Consequently the momentum density of any mol ule will have inversion symmetry, even if p(r) does not. It can be useful to take advantage of this inversion symmetry when integrating functions of p(p such as the generalised overlaps to be described in Sect. 3. 

The bound electron-hole system is known as an exciton It can be thought of as a quasiatom , formed from the particle and the hole, with the two objects rotating about a common centre of mass, the angular momentum being quantised. Just as, in atomic physics, one generalises by considering the electron in a many-electron atom as a quasiparticle, we can now replace the positive centre of charge by a hole. This system is often... 

He also emphasises that confusion between process rates and rates of change must be avoided if an accurate mathematical assessment of the problem is to be achieved. The process rates are those that can be directly related to system variables such as temperature, pressure, composition, velocity and geometry (e.g. flow area). These fundamental quantities involve conservation of mass, energy, momentum and chemical species and may be generalised and simply correlated. Rates of change on the other hand, cannot be simply correlated or generalised, and involve the rate of accumulation of the biofilm and the net rate of input by virtue of the flowing system. 

To understand the origin of this power of 3 in a more fundamental way, let us consider the generalisation of our analysis to a hydrogenic wavefunction of angular momentum / in I spatial dimensions. For simplicity, we take the principal quantum number n = / + 1. Then such a hydrogenic eigenfunction has in configuration space the form... 

Introducing the canonical momentum p = Mu, the motion of the system under the applied generalised forces Q is described by the DAE system of index 2... 

Prom these quantities it is seen that Ujj = 0, because a and c have no spatial dependence. Further, the material time derivative of the velocity satisfies = 0. In the absence of any external body force Fi and generalised body forces Gf and GJ, the balance of linear momentum (6.211) consequently reduces to... 

---