Commutation orbital angular momentum

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

The difference between the two solutions with the same sign of energy can only be in their spin states. Since plane waves have no orbital angular momentum, the spin commutes with the Hamiltonian, and since ct3Ui = Mi but ct3m2 = — u2, the two solutions correspond to spin states with Sz = h/2 respectively. 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

Any component of S commutes with any component of L and, as for orbital angular momentum, it is readily shown that... 

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. 

Because V depends only on r, one finds that this Hamiltonian commutes with the orbital angular-momentum operators L2 and Lz. Hence the... 

The molecular electronic wave functions ipe] are classified using the operators that commute with Hei. For diatomic (and linear polyatomic) molecules, the operator Lz for the component of the total electronic orbital angular momentum along the internuclear axis commutes with Hel (although L2 does not commute with tfel). The Lz eigenvalues are MLh,... 

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. 

The commutation relations of the orbital angular momentum operators can be derived from those between the components of r and p. If we denote the Cartesian components by the subindices i, k, and /, we can use the short-hand notation... 

By contrast, the square modulus of the orbital angular momentum (see Eq. [22]) commutes with all three components of S , that is,... 

As for the orbital angular momentum, the commutation relations between the Cartesian components of a general angular momentum / and its square modulus A read... 

We saw in section 3.3 that neither the orbital (TiL = R a P) nor the spin (1/2)7/a angular momenta commute with the Dirac Hamiltonian and are not therefore separate constants of motion, although their sum is. However, we can construct the mean orbital angular momentum and mean spin angular momentum operators in the same way as in equation (3.129). These operators are, respectively,... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Conserved operators commute with the Hamiltonian. We want to know if the orbital angular momentum L (3.59) is conserved. Consider Lx, using the commutation relations (3.6) and equn. (3.161). 

The angular part of the Laplacian corresponds to the square of the orbital angular momentum operator l2. Its second term includes the square of its z-component lz. Both operators commute with each other, and each one commutes with the Hamiltonian for any central potential, including the... 

Orbital hybridization, like the Bohr model of the hydrogen atom in its ground state, is an effort to dress up a defective classical model by the assumption ad hoc of quantum features. The effort fails in both cases because the quantum-mechanics of angular momentum is applied incorrectly. The Bohr model assumes a unit of quantized angular momentum for the electron which is presumed to orbit the nucleus in a classical sense. Quantum-mechanically however, it has no orbital angular momentum. The hybridization model, in turn, spurns the commutation rules of quantized... 

Applied to a central field, the theory may be tested for the constancy of the orbital angular momentum of the charged particle. It is found that the operator representing this quantity does not commute with the Hamiltonian operator, i.6o that the orbital angular momentum is not a constant of the motion. A quantity may be constructed which will commute with the Hamiltonian operator by adding to the orbital... 

We note that the (each component of the) spin S commutes with the orbital angular momentum and hence... 

Of those five operators only three can each commute with the others (say s2,sz,H0) and thus only these operators may have the same set of wave functions s,ms,E >. Unlike the orbital angular momentum, neither is the differential form of the spin operators presented, nor is the functional form of the spin functions known. 

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... 

The classical-mechanical definition of orbital angular momentum is L = r x p. The operator commutes with L, Ly, and but L , Ly, and do not commute with one another. When expressed in spherical coordinates, the operators L, L Ly, and depend only on the angles 0 (the angle between the z axis and r) and (the angle between the projection of r in the xy plane and the x axis) and not on the radial coordinate r. 

We postulate that the spin angular-momentum operators obey the same commutation relations as the orbital angular-momentum operators. Analogous to [Lj, Ly = ihL [Ly, LJ = ihL [Lj, Lj,] = ihLy [Eqs. (5.46) and (5.48)], we have... 

Although the individual orbitd-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) is the square of the magnitude of the toted electronic orbital angular momentum. The electronic wave function il/ of an atom satisfies L tfr = L(L -I- The total-electronic-orbital-angular-momentum quantum number L of an atom is specified by a code letter, as follows ... 

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. 

For Hz, the operator L commutes with H. For a many-electron diatomic molecule, one finds that the operator for the axial component of the total electronic orbital angular momentum commutes with H. The component of electronic orbital angular momentum along the molecular axis has the possible values Mjh, where = 0, 1,... 

For nonlinear polyatomic molecules, no orbital angular-momentum operator commutes with the electronic Hamiltonian, and the angular-momentum classification of electronic terms cannot be used. Operators that do commute with the electronic Hamiltonian are the symmetry operators Or of the molecule (Section 12.1), and the electronic states of polyatomic molecules are classified according to the behavior of the electronic wave function on application of these operators. Consider H2O as an example. 

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