Cartesian coordinates momentum

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

The third quantum number m is called the magnetic quantum number for it is only in an applied magnetic field that it is possible to define a direction within the atom with respect to which the orbital can be directed. In general, the magnetic quantum number can take up 2/ + 1 values (i.e. 0, 1,. .., /) thus an s electron (which is spherically symmetrical and has zero orbital angular momentum) can have only one orientation, but a p electron can have three (frequently chosen to be the jc, y, and z directions in Cartesian coordinates). Likewise there are five possibilities for d orbitals and seven for f orbitals. 

The form of Lz in Cartesian coordinates is Eq. (7-la), and it is clear that orbital angular momentum is related to angular displacement in the same way as the linear operators are related. 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

For illustration, we consider some examples involving only one variable, namely, the cartesian coordinate x, for which w x) = 1. An operator that results in multiplying by a real function /(x) is hermitian, since in this case fix) = fix) and equation (3.8) is an identity. Likewise, the momentum operator p = (i)/i)(d/dx), which was introduced in Section 2.3, is hermitian since... 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

Consider the fluid s x-component of motion in a rectangular Cartesian coordinate system. By following the flow, the rate of change of a fluid element s momentum is given by the substantive derivative of the momentum. By Newton s second law of motion, this can be equated to the net force acting on the element. For an element of fluid having volume Sx ySz, the equation of motion can be written for the x-component as follows ... 

Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

In the special case that the scattering vector is parallel to one of the coordinate axes, these expressions look much simpler. For example, if q is parallel to the z axis, the directional Compton profile, expressed in Cartesian coordinates, is simply the marginal momentum density along the axis ... 

For most molecules, the small momentum expansion of the momentum density requires the full 3x3 Hessian matrix A of n( p) at p = 0. In Cartesian coordinates, this matrix has elements... 

Notice that H is a second order differential operator in the space of the thirty-nine cartesian coordinates that describe the positions of the ten electrons and three nuclei. It is a second order operator because the momenta appear in the kinetic energy as pj2 and pa2, and the quantum mechanical operator for each momentum p = -ih dfdq is of first order. 

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... 

Here N is the extensive variable associated with the conservation law (e.g., the momentum vector P), p is the fluid s mass density, and t] is the intensive variable associated with N (e.g., the velocity vector V). The volume of the control volume is given as 6V. In a cartesian coordinate system (, y, z),SV = dxdydz. The operator D/Dt is called the substantial derivative. 

Balance Equations on a Differential Control Volume When the net forces are substituted into Eq. 2.14, the 8 V cancels from each term, leaving a differential equation. As a very brief illustration, a one-dimensional momentum equation in cartesian coordinates is written as... 

A sound understanding of the physical conservation laws is essential to one s ability to specialize them, solve them, and apply the results successfully. Therefore we begin with a derivation of the laws that govern the conservation of mass, momentum, thermal energy, and chemical species. We approach the derivation from a fluid-mechanical point of view, and the reader may find considerable overlap with other books in viscous fluid mechanics. However, we depart from the traditional presentation in two ways. First, because we are principally concerned with chemically reacting flow, we retain many features that may be negligible in fluid flow alone. Second, because we are often concerned with axisymmetric flows, we cast much of the mathematics in cylindrical coordinates rather than cartesian coordinates. While the later choice adds some complexity, it also serves to highlight some important issues that can be overlooked in cartesian coordinates. 

Consider a planar collision of a sphere of mass m and radius a against an initially stationary sphere of mass m2 and radius a2. Select the Cartesian coordinates so that the x- and y-axes are normal and tangential to the contact surfaces, respectively, as shown in Fig. 2.2. The conservation of linear momentum yields... 

Consider a cubic unit volume containing n particles in a Cartesian coordinate system. On average, about n/6 particles move in the +y-direction, and the same number of particles move in the other five directions. Each particle stream has the same averaged velocity (v). Since particle collision is responsible for the momentum transport, the averaged x-component of the particle momentum transported in the y-direction may be reasonably estimated by... 

Consider the propagation of a one-dimensional normal shock wave in a gas medium heavily laden with particles. Select Cartesian coordinates attached to the shock front so that the shock front becomes stationary. The changes of velocities, temperatures, and pressures of gas and particle phases across the normal shock wave are schematically illustrated in Fig. 6.12, where the subscripts 1, 2, and oo represent the conditions in front of, immediately behind, and far away behind the shock wave front, respectively. As shown in Fig. 6.12, a nonequilibrium condition between particles and the gas exists immediately behind the shock front. Apparently, because of the finite rate of momentum transfer and heat transfer between the gas and the particles, a relaxation distance is required for the particles to gain a new equilibrium with the gas. 

In quantum mechanics, the independent variables q and p of classical mechanics are represented by the Hermitian operators q and p with the following matrix elements in the Cartesian coordinate basis q), here just written for the coordinate g and the conjugate momentum pp... 

We note that volumes in phase space have units of kg m2/s = J s for each Cartesian coordinate. In order to have a dimensionless degeneracy, g, we divide volumes in phase space, dr dp, by a constant, h2, for which h has units of J s. In other words, g(r, p) = h independent of position and momentum ... 

The connection between the covariant cyclic and cartesian coordinates of the vector J yields Eq. (A.6), whilst (A.5) makes it possible to form the vector itself out of the components (J)q. As follows from (2.18), the components of the multipole moment pq characterize the preferred orientation of the angular momentum J in the molecular ensemble. Fig. 2.3(a, b) shows the probability density p(0, [Pg.30]

This result is interpreted to mean that angular momentum is described by the operator L(y ) = ihd/dp, which is equivalent to the postulate that linear momentum is quantum-mechanically represented by the operator —ihV. The Bohr operator for angular momentum is converted into cartesian coordinates by writing... 

The four terms in 3C , however, require closer attention. The second term, which in molecule-fixed cartesian coordinates may be expressed as L2 + L2, affects all levels equally and is therefore usually omitted. The third term, which does not involve the orbital angular momentum, is known as the spin uncoupling term. Its matrix elements... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

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