Spin-waves

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

In general, for a homonuclear diatomic molecule there are (21+ )(/+1) symmetric and (21+ 1)/antisymmetric nuclear spin wave functions therefore... 

If / = 1 for each nucleus, as in H2 and N2, the total wave function must be symmetric to nuclear exchange. There are nine nuclear spin wave functions of which six are symmetric and three antisymmetric to exchange. Figure 5. f 8 illustrates the fact that ortho- ll2 (or N2)... 

The spin part pl can be derived by labelling the electrons 1 and 2 and remembering that, in general, each can have an a or /i spin wave function giving four possible combinations a(l)P(2), P(l)a(2), a(l)a(2) and P(l)P(2). Because the first two are neither symmetric nor antisymmetric to the exchange of electrons, which is equivalent to the exchange of the labels 1 and 2, they must be replaced by linear combinations giving... 

For helium, therefore, the singlet spin wave function of Equation (7.24) can combine only with the orbital wave function of Equation (7.26) giving, for singlet states. 

Equation (7.23) expresses the total electronic wave function as the product of the orbital and spin parts. Since J/g must be antisymmetric to electron exchange the Ig and Ag orbital wave functions of oxygen combine only with the antisymmetric (singlet) spin wave function which is the same as that in Equation (7.24) for helium. Similarly, the Ig orbital wave function combines only with the three symmetric (triplet) spin wave functions which are the same as those in Equation (7.25) for helium. 

The computation could also be made into a ballistic action by adding sites to the left and right of the sites to be used for the actual computation (with appropriate ff i s but with no corresponding Ai). Instead of starting the computations by putting a cursor at site 0, the computation can be initialized with an incoming packet of some specified momentum (i.e. a spin wave) by putting the cursor with different amplitudes at different sites. 

Let us further restrict ourselves to the case of K = 0 spin wave states, namely, those states that can be sampled usually by electromagnetic waves. Hence, then, it is necessary to have the character... 

Finally, this type of analysis can be carried out for any point in the Brillouin zone such that by using the transformation properties of spin waves and the character tables, one may obtain the spin-wave band structure throughout the zone. 

Spin waves, 753 Spin wave states, 757 Spin zero particles, 498 "Spinor space, 428 Spinors, 394,428, 524 column, 524 Dirac... 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

Considering first the state with a total spin of 0, we note that since the spin wave function is antisymmetric with respect to interchanging the particle labels, the spatial part of the wave function should be symmetric in order to preserve the overall antisymmetry of the wave function. This leads to the following expression for the wave function ... 

For the triplet spin states it is only necessary to consider one of the three possible spin wave functions and we will take the J wave function. Since... 

Numerous experimental studies have shown that the magnetic hyperfine field of magnetic nanoparticles varies linearly with temperature at low temperatures, in accordance with (6.23). This is in contrast to bulk materials for which the decrease in the hyperfine field with increasing temperature in accordance with spin wave... 

To aid our understanding of absorption and emission processes, Eq. (2.1) can be expanded in terms of electronic, vibronic (vibrational components of an electronic transition), and spin wave functions ... 

If any one of these integrals (expectation value equations) is zero, the transition is said to be forbidden. For the electronic and spin wave functions, it is not necessary to evaluate the integral but only to note that an odd function integrated from minus infinity to infinity is zero, while an even function integrated within these limits results in a nonzero value. For example (Figure 2.1),... 

It would be important to figure out the low energy excitation modes (Nambu-Goldstone modes) built on the ferromagnetic phase. The spin waves are well known in the Heisenberg model [10]. Then, how about our case [32] ... 

As a consequence of the quantum-mechanical selection rules for resonance energy transfer involving the spin wave functions of the donor and acceptor,... 

J = 1,3,5 — are antisymmetric with respect to the nuclear coordinates. It follows that homonuclear diatomic molecules with anti-symmetric nuclear spin wave functions (nuclei with half-integer I = 1/2, 3/2...) can combine only with symmetric rotational functions (even J = 0,2,4...), while those with symmetric nuclear spin wave functions (even I) can combine only with antisymmetric rotational functions... 

For each of the diatomic examples above, examples which include all possible combinations of symmetric or anti-symmetric nuclear spin wave functions... 

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