Zeeman effect Hamiltonian, spin

Abstract. Following a suggestion of Kostelecky et al. we have evaluated a test of CPT and Lorentz invariance from the microwave spectrosopy of muonium. Precise measurements have been reported for the transition frequencies U12 and 1/34 for ground state muonium in a magnetic field H of 1.7 T, both of which involve principally muon spin flip. These frequencies depend on both the hyperfine interaction and Zeeman effect. Hamiltonian terms beyond the standard model which violate CPT and Lorentz invariance would contribute shifts <5 12 and <5 34. The nonstandard theory indicates that P12 and 34 should oscillate with the earth s sidereal frequency and that 5v 2 and <5 34 would be anticorrelated. We find no time dependence in m2 — vza at the level of 20 Hz, which is used to set an upper limit on the size of CPT and Lorentz violating parameters. 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

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

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

The simplest possible description of the Zeeman interaction is a single-term effective Hamiltonian describing the magnetic interaction between the applied field and the electron spin magnetic moment,... 

The effective Hamiltonian used by Cecchi and Ramsey [63] to analyse the strong magnetic field spectrum was the sum of four terms, describing the molecular rotation, nuclear spin interactions, Stark interactions and Zeeman interactions. Specifically the Hamiltonian is the following,... 

The effective Hamiltonian used by Saykally, Evenson, Comben and Brown [62] contains terms which we have already met in this chapter, and which we will therefore deal with fairly briefly, with appropriate references to the details given elsewhere, particularly in this section. The theory has been developed in a number of papers, particularly by Brown, Kopp, Malmberg and Rydh [63], Brown and Merer [64], who dealt with n states of triplet and higher spin multiplicity, and Steimle and Brown [65] who specifically addressed the theory of the A-doubling of CO in the 3 n state. The theory of the Zeeman interactions follows closely that developed to analyse the magnetic resonance spectra of OH by Brown, Raise, Kerr and Milton [66]. All of these... 

We have chosen to use the hyperfine-coupled representation, where for 12CH, F is equal to J 1 /2. An appropriate basis set is therefore t], A N, A S, J, /, F), with MF also important when discussing Zeeman effects. As usual the effective zero-field Hamiltonian will be, at the least, a sum of terms representing the spin-orbit coupling, rigid body rotation, electron spin-rotation coupling and nuclear hyperfine interactions, i.e. 

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