Momentum mechanical

The device of Figure 2-70 can also operate as a motor if a DC current is applied to the rotor windings as in the alternator and an AC current is imposed on the stator windings. As the current to the stator flows in one direction, the torque developed on the rotor causes it to turn until the rotor and stator fields are aligned (5 = 0°). If, at that instant, the stator current switches direction, then mechanical momentum will carry the rotor past the point of field alignment, and the opposite direction of the stator field will cause a torque in the same direction and continue the rotation. 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

Fluid flow, also known as fluid mechanics, momentum transfer and momentum transport, is a wide-ranging subject, fundamental to many aspects of chemical engineering. It is impossible to cover the whole field here, so we will concentrate upon a few aspects of the subject. 

As photon momentum p = E/c, the quantum assumption E = hu implies that p = hu/c = h/X. This relationship between mechanical momentum and wavelength is an example of electromagnetic wave-particle duality. It reduces the Compton equation into ... 

Momentum quantity transferred by turbulent mechanism Momentum quantity transferred by molecular mechanism... 

The electronic magnetic multipoles (25)-(27) are unperturbed, or permanent, moment operators. In the presence of a vector potential A(r, t) (we simplify the notation, omitting the index), the canonical momentum is replaced by the mechanical momentum... 

We now shift our focus in the free electron model to the momentum operator. According to Table 3.2, the quantum mechanical momentum operator in one-dimension is given by Equation (11.11). 

It must be recalled that, in the Formal Graph approach, the notion of conductor is generalized to any device or material possessing the constitutive property of conductance. This includes the classical concept of conductor of particles (charges, molecules, etc.) but also conductors of entities that are momenta, impulses, volumes, lengths, and so on. Friction, for instance, is a conduction process of mechanical momentum in a viscous fluid or of geometric entities (surfaces, etc.) in a solid (in that case one speaks of internal friction between solid elements). 

The same symbols will be used for classical quantities and also for the corresponding operators of the quantized formulation. Consequently, p might s)mibolize classical momentum as well as the quantum mechanical momentum operator p = —ihV for example. The detailed meaning of symbols will become obvious from the context. Occasionally one might encounter a hat on top of a s)mibol chosen in order to emphasize that this symbol denotes an operator. However, a hat on top of a vector may also denote the corresponding unit vector pointing in the direction of the vector, e.g., the position vector may therefore be expressed asr = rr. 

Another interesting idea to generahze the adiabatic states arises from Eq. (2.9). By replacing the quantmn mechanical momentum operators for nuclei (Pfc) with their classical coimterparts (P/t), one obtains another electronic Hamiltonian... 

It may seem disconcerting that p involves imaginary numbers, because the momentum of a free particle is a real, measurable quantity. The quantum mechanical momentum given by ( P p P), however, also proves to be a real number (Box 2.1). The formula for p emerged from the realization by Max Bom, Werner Heisenberg, Paul Dirac and others in the period 1925-1927 that the momentum of a botmd particle such as an electron in an atom cannot be specified precisely as a function of the particle s position. 

From classical mechanics (e.g., Goldstein 1950), we can show that the presence of a vector potential requires that the Hamiltonian function must be constructed using the kinetic momentum (or mechanical momentum), which is the momentum that is given in nonrelativistic theory by m. We must express this momentum in terms of the canonical momentum of Lagrangian mechanics, because it is the canonical momentum to which the quantization rule p —ihV applies. Here (and hereafter) we will use p for the canonical momentum and n for the kinetic momentum. The relation between the two is... 

For a classical expression of this kind, we turn to the relation between energy and momentum derived in section 2.7. Omitting the subscript for the rest mass of the electron and introducing the mechanical momentum, we may write this relation as... 

What is the significance of the four components We noted above that in nonrelativistic quantum mechanics we can introduce the spin by replacing the mechanical momentum nr by a nr and making the wave function a two-component vector, or 2-spinor, where the upper component corresponds to spin j and the lower component corresponds to spin -i. The same concept applies to the Dirac wave function— components 1 and 3 correspond to spin and components 2 and 4 correspond to spin — i, and the wave function is called a 4-spinor. 

We should note that in the presence of electromagnetic fields the canonical momentum p is no longer equal to the product of mass and velocity. The latter is therefore often also called the kinematical or mechanical momentum if... 

However, in order to use the usual substitution rule, Eq. (2.45), for the transition to quantum mechanics, the classical Hamiltonian has to be written in terms of the canonical momentum p, and not the velocity v. But with the help of Eq. (2.57) we can replace the mechanical momentum by the canonical momentum and get... 

In the previous sections it was shown that in the minimal coupling approximation the vector potential enters the mechanical momentum of electron i... 

Equation (5.7) gives a general expression for the radiation force on an atom moving in a laser field. From a quantum mechanical point of view, the radiation force (5.7) arises as a result of the quantum mechanical momentum exchange between the atom and the laser field in the presence of spontaneous relaxation. The change in the atomic momentum comes from the elementary processes of photon absorption and emission stimulated absorption, stimulated emission, and spontaneous emission. The radiation force (5.7) is a function of the coordinates and velocity of the center of mass of the atom. 

We have omitted in these equations all the terms dependent on the (inverse) masses of the nuclei. In the purely electronic and purely nuclear parts we have not included the rest mass of the particles. In the above equations we have introduced the mechanical momentum... 

Some of the terms included in the Breit-Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spin-orbit interaction (O Eqs. 11.13 and O 11.14) becomes important The Breit-Pauli Hamiltonian in O Eqs. 11.5-11.22, however, only includes molecule-external field interactions through the presence of a scalar electrostatic potential 0 (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (O Eq. 11.23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials. 

We consider a molecule in which the magnetic field arises from two primary sources, an external magnetic field induction and the permanent magnetic moments of nuclei possessing a spin. In the minimal coupling approximation, the mechanical momentum operator (O Eq. 11.23) is given as... 

Balance equations for momentum, mass, and energy transfer provide the broad foundation for much of the physical problems we encounter in chemical engineering curriculum. They represent important starting steps for developing phenomenological events especially in dealing with fluid mechanics (momentum transfer), heat and mass transfer, and reaction studies as well. 

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