Total orbital angular momentum operator

L the total orbital angular momentum operator the total spin angular momentum operator... 

After separating the quadratic term into expressions containing individually the total orbital angular momentum operator L and the total spin angular momentum operator S, it can be shown that [55]... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

The eigenvalues and eigenfunctions of the orbital angular momentum operators can also be derived solely on the basis their commutation relations. This derivability is particularly attractive because the spin operators and the total angular momentum obey the same commutation relations. 

Stone applied similar reasoning to the problem of a three-dimensional cluster. Here, the solutions of the corresponding free-particle problem for an electron-on-a-sphere are spherical harmonics. These functions should be familiar because they also describe the angular properties of atomic orbitals.Two quantum numbers, L and M, are associated with the spherical harmonics, Yim 0,total orbital angular momentum and its projection on the z-axis, respectively. It is more convenient to use the real linear combinations of Yim 9,(p)dinA (except when M = 0), and... 

Other two-particle operators can also be expressed in terms of the general pair creators and annihilators. For instance, the total orbital angular momentum is... 

A model hamiltonian should have the structure of the full hamiltonian, but could in principle have terms consisting of higher order products of annihilation and creation operators. Here we limit considerations to such operators that contain a one-electron part and an electron-electron interaction part. The number of independent matrix elements can be considerably reduced by symmetry considerations and by requiring compatibility with other operator representatives. It is clear that the form of the spectral density requires that the hamiltonian commutes with the total orbital angular momentum and with various spin operators. These are given in the limited basis as... 

Analogous to the orbital angular-momentum operators L, L L L, we have the spin angular-momentum operators S, S, Sy, S which are postulated to be linear and Hermitian. is the operator for the square of the magnitude of the total spin angular momentum of a particle. is the operator for the z component of the particle s spin angular momentum. We have... 

We have based the discussion on a scheme in which we first added the individual electronic orbital angular momenta to form a total-orbital-angular-momentum vector and did the same for the spins L = S,- L, and S = 2i S,. We then combined L and S to get J. This scheme is called Russell-Saunders couplit (or L-S coupling) and is appropriate where the spin-orbit interaction energy is small compared with the interelec-tronic repulsion energy. The operators L and S commute with + W,ep, but when is included in the Hamiltonian, L and no longer commute with H. (J does commute with + //rep + Q ) If the spin-orbit interaction is small, then L and S almost commute with (t, and L-S coupling is valid. 

The dipole moment mo for an unperturbed system depends on the total angular momentum, which may be written in terms of the orbital angular momentum operator Lg and the total electron spin S. 

COMMENT. As described at the end of Section 8.6, the physicai properties associated w/ith non-commuting operators cannot be simultaneously known with precision. However, since H,P, and Iz commute we may simultaneously have exact knowledge of the energy, the total orbital angular momentum, and the projection of the orbital angular momentum along an arbitrary axis. 

Although the individual orbital-angular-momentum operators L, do not commute with the atomic Hamiltonian (11.1), one can show (Bethe and Jackiw, pp. 102-103) that L does commute with the atomic Hamiltonian [provided spin-orbit interaction (Section 11.6) is neglected]. We can therefore characterize an atomic state by a quantum number L, where L(L -I- 1 )tt is the square of the magnitude of the total electronic orbital angular momentum. The electronic wave function of an atom satisfies = L L + 1 )h tl/. The total-electronic-orbital-angular-momentum quantum number L of an atom is specified by a code letter, as follows ... 

The iV-electron operator fh RGo) is called the orbital magnetic dipole operator, kiRoo) is the orbital angular momentum operator of electron i with respect to the gauge origin Rqo and Sj is the spin angular momentum operator of electron i. The total orbital and spin angular moment operators of all N electrons are... 

In this case, then, the orbital degeneracy can be described by introducing into the spin Hamiltonian the operator for the unquenched part of the total orbital angular momentum. The term y4L S2 is the analogue of the commonly adopted free-atom form fL S, which is appropriate in a strictly central-field situation. Again, however, the effective Hamiltonian corresponds to a simple model, all the complexities of the real system... 

Applying the tensor properties of the electric dipole operator and the Wigner-Eckart theorem to the space of the total orbital angular momentum (L), we derive the selection rules... 

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