Angular Momentum Subspace

This is a function in the so-called partial wave subspace (or angular momentum subspace) L ((0,oo),dr) fCmj,Kj- The norm of this function is given by... 

For example, the action of K is just multiplication by the eigenvalue —Kj. The action of the Dirac matrices / and a in the partial wave subspace is described by (110). Likewise, we can compute the action of a spherically symmetric potential in one of the angular momentum subspaces. It remains to observe that due to the factor r in (102) the operator djdr - 1/r in (which is part of expression for the Dirac operator in polar coordinates) simply becomes d/dr in L (0,oo) ... 

Fig. 20. A resonance state for the H + HD system at Ec = 1.2 eV at a total angular momentum of J = 25. In (a) the wavefunction is shown in the H + HD Jacobi coordinates for the collinear subspace. In (b) the wavefunction has been sliced perpendicular to the minimum energy path and is plotted in symmetric stretch and bend normal mode coordinates.

Consider, furthermore, a (2i- - 1)-dimensional subspace of the Hilbert space with fixed 5. Then, according to Schwinger s theory of angular momentum [98], this discrete spin DoF can be represented by two bosonic oscillators described by creation and annihilation operators with commutation relations... 

Some readers may wonder why we make this restriction, especially if they have experience applying angnlar momentnm operators to discontinuous physical quantities. It is possible, with some effort, to make mathematical sense of the angular momentum of a discontinuous quantity hut, as the purposes of the text do not require the result, we choose not to make the effort. Compare spherical harmonics, which are effective because physicists know how to extrapolate from spherical harmonics to many cases of interest by taking linear combinations likewise, dense subspaces are useful because mathematicians know how to extrapolate from dense subspaces to the desired spaces. 

But the difficulty can be overcome if we observe that Il0, 11° and H(i) are all invariant under the rotation about the z-axis and that in consequence we can consider the problem in each of the subspaces of where the z-component of the angular momentum takes on definite values. Considered in any one fm) of such subspaces belonging to the magnetic quantum number m, Il0) reduces to a constant, so that we have only to take the term k II into consideration. Then, since H0 and If"J are both bounded below, we can apply the Case ii) of 7. 3 and conclude that the condition C) is also satisfied in... 

Since we are interested in the case with zero total angular momentum, we have chosen such initial conditions for three particles. However, the phase space of the entire initial conditions is too big— in fact, infinite. Therefore, for numerical investigation, we have to restrict the initial conditions to some subspace of the entire initial conditions. If three particles have zero velocity at some moment, then the orbit associated to this initial condition has zero angular momentum, since L = i Qi Pi- Thus we consider the initial conditions with zero velocities of three particles at time zero. These are just the initial conditions of the free-fall. So this problem is sometimes called the free-fall problem. The free-fall problem in gravitational three-body problem was well investigated [35-38]. 

We consider the subspace of iV-electron states with the same total angular momentum and parity, denoted respectively by j and Since these states are orthogonal to states with different j, the subspaces are independent and the problems of the interaction of states are separate. 

We consider the subspace of radial states belonging to the angular-momentum index f, which we drop from the notation for the states. u r) is the coordinate representation of a state n). 

In particular, the Dirac operator leaves each of the partial wave subspaces invariant, that is, the result of its action is a wave function in the same angular momentum wave subspace. 

The Wigner matrices multiply just like the rotations themselves. There is a one-to-one correspondence between the Wigner matrices of index l and the rotations R. These matrices form a representation of the rotation group. In fact, since the 2/ + 1 spherical harmonics of order / form an invariant subspace of Hilbert space with respect to all rotations, it follows that the matrices D ,m(R) form a (21 + 1) dimensional irreducible representation of the rotation R. Explicit formulas for these matrices can be found in books on angular momentum (notably Edmunds, 1957). 

If the decay of the excited state takes place, before the spin is coupled to the orbital angular momentum the electron distribution is determined by Phg L,t = 0). If, on the other hand, the lifetime is sufficiently large, then the spin is coupled and the anisotropy, which initially was only present in the L-subspace, is partly transferred into the spin subspace, where it does not influence the angular electron distribution. The time-dependent tensor components Pkq(U t) are then given by (see, e.g., the treatment of Blum, Chapter 4.7.2)... 

Besides the magnetic dipole moment, nuclei with spin higher than 1/2 also possess an electric quadrupole moment. In a semiclassical picture, the nuclear electric quadrupole moment informs about the deviation of nuclear charge distribution from spherical symmetry. Nuclei with spin 0 or 1/2 are therefore said to be spherical, with zero electric quadrupole moment. On the other hand, if the nuclear spin is higher than 1/2, the nuclei are not spherical, assuming cylindri-cally symmetrical shapes around the symmetry axis defined by the nuclear total angular momentum [17]. Within the subspace [/,m), the nuclear electric quadrupole moment operator is a traceless tensor operator of second rank, with Cartesian components written is terms of the nuclear spin [2] ... 

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