Hamiltonian operator Zeeman effect

Example 6. Zeeman effect of the hydyogen-like atom (spin neglected). The Hamiltonian operator is given by... 

The prerequisite for the creation of orientation in the aligned state can also be formulated in terms of the time reversal properties of a Hamiltonian operator which represents the perturbation. As is shown in [276, 277] the alignment-orientation conversion may only take place if the time invariant Hamiltonian is involved. For instance, the Hamiltonian operator of the linear Zeeman effect is odd under time reversal and is thus not able to effect the conversion, whilst the operator of the quadratic Stark effect is even under time reversal and, as a consequence, the quadratic Stark effect can produce alignment-orientation conversion. 

This table contains the data obtained from the magnetic hyperfine structure and the Zeeman effect for molecules in a Z state or more generally in a state with Q = 0, i.e., the projection of the angular momentum onto the molecular axis is zero. For the magnetic hyperfine stracture one usually considers four terms the spin-rotation interaction for each nucleus and the scalar and tensorial spin-spin interaction of the two nuclear spins. For the Zeeman effect one takes into account the rotational Zeeman effect, the nuclear Zeeman effect with the scalar and tensorial shielding, and the scalar and tensorial magnetic susceptibility. The hamiltonian of these interactions can be written with the concept of spherical tensor operators [57Edm]... 

An applied field in the xy-plane can tune the tunnel splittings Amm. via the Sx and Sy spin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This effect can be demonstrated by using the Landau-Zener method (Section 3.1). Fig. 8 presents a detailed study of the tunnel splitting A 10 at the tunnel transition between m - +10, as a function of transverse fields applied at different angles (p, defined as the azimuth angle between the anisotropy hard axis and the transverse field (Fig. 4). 

The effective Zeeman Hamiltonian for an asymmetric top molecule in an open-shell state has been given by several authors in the literature, for example Pryce [50Pry] or Bowater, Brown, and Carrington [73 Bow], The operator can be written in Cartesian tensor notation as [80Eve] ... 

To evaluate the effeet of spin-orbit coupling we will start writing down the first-order corrected wave functions of the Ms = 5 sublevels, then calculate the matrix elements of the Zeeman Hamiltonian (Eq.2.23) and compare these to the matrix elements of the (effective) spin-only Zeeman Hamiltonian given in Eq. 2.48 to find analytical expressions for the diagonal elements of g. With fZ, 5 as perturbation operator, the wave functions that describe the lowest two levels become... 

In the cases where first-order calculation is appropriate, the effects of the Hamiltonian Hq given in Equation (2.7.22) can be readily computed as both operators 1 and commute with the main Hamiltonian Hq. The result is that the energy levels are not equally spaced as they were in the case of the Zeeman interaction only. Otherwise, the energy depends on the quantum number m and the parameter (oq in the form ... 

For an ensemble of nuclei with spin / > 1/2 experiencing no quadrupolar interaction, such as in an isotropic liquid or a crystal with cubic symmetry, the equations describing the time evolution of the density matrix under action of static and RF magnetic fields are a natural extension of the / = 1/2 case. The Hamiltonian contains only the Zeeman and RF terms the effects of RF pulses are described by rotations of the spin operators around the transverse axes in the rotating frame, whereas free evolution corresponds to rotations around the z-axis. 

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