Hamiltonian systems angular momentum

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 ... 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

The presence of two angular momenta has as a consequence that only their sum, representing the total angular momentum in the case considered, necessary commutes with the Hamiltonian of the system. Thus only the quantum number K, associated with the sum, N, of and Lj,... 

In a many-eleetron system, one must eombine the spin funetions of the individual eleetrons to generate eigenfunetions of the total Sz = i Sz(i) ( expressions for Sx = i Sx(i) and Sy =Zi Sy(i) also follow from the faet that the total angular momentum of a eolleetion of partieles is the sum of the angular momenta, eomponent-by-eomponent, of the individual angular momenta) and total S2 operators beeause only these operators eommute with the full Hamiltonian, H, and with the permutation operators Pij. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not eommute with Pij. 

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. 

Accordingly, the quantum-mechanical Hamiltonian operator H for this system is proportional to the square of the angular momentum operator U-... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The vibrational dynamics of this system can be adequately studied by a two degrees of freedom model, with the C-N distance kept frozen at its equilibrium value of re = 2.186 a.u. The vibrational (total angular momentum J = 0) Hamiltonian in scattering or Jacobi coordinates is given by... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

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. 

Figure 8.1 Schematic structure of the Hamiltonian matrix for a collision system in an external field expressed in the total angular momentum representation.

The complete group of an electronic system is defined by the set of all unitary operators which commute with the hamiltonian describing it. The hamiltonian He(p cc) musthave symmetryproperties, thereby introducing constrains in the ct-coordinate space. Operators considered here are those of angular momentum, those which transform the spatial coordinates ofthe system (including Coulomb sources), and those permuting equivalent sites within the system [9] ct may play the role of a symmetry label. 

The quantum states of the molecular hamiltonian can be labelled with the linear momentum of each rigid configuration for the sources of static Coulomb potential, Pi and P2 or koi = fl and k02 = P2/, respectively. The total angular momentum of each cluster, Ji and J2 and their projections Jiz and J2z together with the vibrational and electronic quantum numbers provide a set of labels helping to characterize the quantum states of the system ... 

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