Angular-momentum

Angular momentum plays an important role in both classical and quantum mechanics. In isolated classical systems the total angular momentum is a constant of motion. In quantum systems the angular momentum is important in studies of atomic, molecular, and nuclear structure and spectra and in studies of spin in elementary particles and in magnetism. 

For a fixed mass, the conservation of linear momentum is equivalent to Newton s second law  

For a flow system, streams with curved streamlines may carry angular momentum into and/or out of the system by convection. To account for this, the general macroscopic angular momentum balance applies  

For a steady-state system with only one inlet and one outlet stream, this becomes 

This is known as the Euler turbine equation, because it applies directly to turbines and all rotating fluid machinery. We will find it useful later in the analysis of the performance of centrifugal pumps. 

Also for the total angular momentum Ltot = Lmat + Eem a conservation theorem for closed systems may be found by similar considerations. The total angular momentum of the electromagnetic field, Lem/ is obtained by 

The close analogy of this relation for fields to the relation of angular momentum and linear momentum for particles, f at = I = r x p,is evident. 

In the above discussion we introduced the idea of angular velocity as the rate of change of the angular coordinate / . To describe the rotational motion of the electron, such angular, rather than linear, quantities can actually be used more widely. 

In Equation (A9.37), r is the position vector for the electron andp its linear momentum tangential to the circle on which it moves around the nucleus the angle between them is always 90° (jt/2 radians), and so the sine of the angle between r and/ is always unity. The product used to define L is a vector cross product, which says that the angular momentum is perpendicular to both the electron coordinate and the linear momentum of the electron. In this classical model, we choose the XY plane to coincide with the circle in which the 

The angular momentum is a constant of the orbital motion of the electron, whereas its linear momentum is constantly changing. The definitions in Equation (A9.37) recognize that what is constant is the magnitude of the momentum and the plane in which the electron moves. Hence, the angular momentum is defined as a vector perpendicular to the plane of motion. 

Equation (A9.32) allows the kinetic energy and magnitude of the angular momentum to be written in terms of the angular velocity  

These equations include the distance r of the electron from the nucleus, because, for the same angular velocity, an electron placed further from the nucleus would be moving faster. 

In Section 1.5 we shall use the wave function (1.17) and the machinery for handling angular momentum (Section 1.4) for the computation of the intensities of spectroscopic transitions. 

For the Kratzer potential, which is quadratic in the variable (r - re)/r, the eigenvalues for / 0 can be obtained in closed form. This is also the case for the potential quadratic in (r-r2/r) (Gol dman et al., 1960). This potential does not, however, tend to a finite value as r — oo. 

In view of the central role played by the angular momentum in molecular physics, it is of interest to review briefly its properties.2 For the relative motion the angular momentum is 

In quantum mechanics, 1 is an operator whose Cartesian components satisfy the commutation relations 

This is, incidentally, the first example of a Lie algebra that we encounter. We will return to it later. Also we have set h = 1 in (1.19) to simplify the notation. 

Typically, moments of inertia will be greater in the transition state than in the reactant, angular momentum is conserved and hence rotational energy, Ej, of the transition state will be less than the rotational energy, Ej, of the reactant. The extent to which the reduction in rotational energy, Ej — Ej, depends upon J will vary according to the nature of the reaction that is the extent to which rotational energy can assist the reaction varies. 

There are a number of possible explanations. One is that the transition states for the IT losses in question are extremely loose and possess little rotational energy (i.e. their moments of inertia are much greater in the transition state than in the reactant and they rotate extremely 

On the other hand, if it is accepted that there are significant centrifugal barriers to the H losses under consideration, questions are raised concerning the interpretation of the ionization efficiency curves [170, 171, 588, 589]. The difficulties associated with the normalisation of these curves has been commented on [171]. It is also possible that the appearance energies derived do not accurately portray the centrifugal barriers due, for example, to tunnelling effects [483, 603]. 

The quantum mechanical treatment of a particle whose classical trajectory is not linear considers a particle possessing non-zero angular momentum. Let rotation be in the x — y plane at a distance R from the rotation axis, z (Fig. 3.1). The moment of inertia 

The angular momentum operator has a much simpler form in polar coordinates (Fig. 3.1)  

Since the particle is spatially constrained on a circular trajectory, [r(( )) must include quantum conditions and the energy must be quantized. The wavefunction can be written as iKcj)) = exp(im(t)) so one easily gets  

Boundary conditions on the wavefunction force m to be an integer (the rotational quantum number). The rotational kinetic energy is quantized as follows  

For an arbitrary rotational motion, the full vector expression of the angular momentum along any direction in space must be considered, with its three components  

The wave-packet structure of the beam affects only the factor 

In fact in a normal experiment we have no knowledge of at all on a scale of positions that are comparable to the position characteristics of an electron—atom system, which are of the order of 10 cm. All we know is that the electrons are in the apparatus, whose scale is of the order 10 cm. We must therefore integrate (3.55) over obtaining the factor 

We may thus consider a beam experiment as a collection of beam experiments, each having an eigenstate of momentum as its initial state and a weight f(p, pm, W )P in the collection. The weight is taken into account in estimating the experimental error. 

Since atoms are strongly affected by the central potential of the nucleus, an important part in electron—atom collision theory is played by states that are invariant under rotations. From the general dynamical principle that invariance under change of a dynamical variable implies a conservation law for the canonically-conjugate variable we expect rotational invariance to imply conservation of angular momentum. Hence angular momentum 

1 Orbital angular momentum The orbital angular momentum observable L is defined by 

The acceleration of the particle by the quantum potential — Wq/m balances the classical acceleration W/m so that the particle remains in a fixed position and is prevented from falling into the nucleus by the outward acceleration due to the quantum potential. 

For degenerate states more than one state with eigenvalue E take the form (9). Both are stationary, but a complex superposition, although again a stationary function, does not have the form (9) and could describe particles in motion [48]. 

Invariance Period Pulsation Dynamic Quantum condition 

It was suggested by Levy-Leblond [50] that a well-defined component of angular momentum should accompany a phenomenon of periodicity a around a rotation axis along z, according to a relationship comparable to (2) and (3), i.e. 

Mathematically m derives as an integer from the requirement that the angular wave stays in phase with itself, i. e. 

The alternative view [705], and the one found in mass spectrometry, is to consider that rotation influences the rate by altering the effective critical energy. Equation (1) can be extended, as in eqn. (2), to include the effect of rotation, viz. 

The force that causes centripetal acceleration is called centripetal force. By Newton s second law 

The linear momentum p = mv = mrisa measure of inertia for a particle moving in a straight line. Accordingly, Newton s second law can be elegantly 

A result applying to angular motion can be obtained by taking the vector product of Newton s law  

Consider a particle of mass m in angular motion with the angular velocity vector (a. The angular momentum can be related to the angular velocity using Eqs. (11.49) and (11.39)  

Tensors are the next member of the hierarchy, that begins with scalars and vectors. The dot product of a tensor with a vector gives another vector. Usually, the moment of inertia is defined for a rigid body, a system of particles with fixed relative coordinates. We must then replace m x + ) in the matrix by the corresponding summation over all particles, mj(y + zj), and so forth. We will focus on the simple case of circular motion, with r perpendicular to o) as in Fig. 11.9. This implies that r = 0, so that the moment of inertia reduces to a scalar  

L is a vector as the result of a cross product and L = iLx+ jLy + kL. Then to form a quantum mechanical operator, one just thinks of the (x,y,z) coordinates as operators and substitutes 

Since we know that H = for the rigid rotor and =, we deduce from 

FIGURE 13.3 Cones of definite angular momentum but indeterminant and Ly for 1 — 2. 

We would have needed nine partial derivatives for V r, 0,4 ) but we only need six to convert to polar coordinates  

Amazingly much of this expression cancels out and leaves only L =- We leave that proof to 

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The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

Klelman V, Gordon R J, Park H and Zare R N 1998 Gompanlon to Angular Momentum (New York Wiley)... 

ZareRN 19SS Angular Momentum (New York Wiley)... 

By multiplying this result by a factor of-2, and adding the result to the conservation of energy equation, one easily finds g = gj = v j - vj. This result, taken together widi conservation of angular momentum, x.gb =... 

Figure A3.4.7. Sunnnary of statistical theories of gas kinetics with emphasis on complex fomiing reactions (m the figure A.M. is the angular momentum, after Quack and Troe [27, 36, 74]). The indices refer to the following references (a) [75, 76 and 77] (b) [78] (c) [79, and M] (d) [31, 31 and M] (e) [, 31 and...

A specific unimolecular rate constant for the decay of a highly excited molecule at energy E and angular momentum J takes the fomr... 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

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