Total state

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

The total state density of redox electrons is then expressed in Eqn. 8-3 ... 

Furthermore, the total state density of redox electrons Z)nEDox(e) in hydrated redox particles near the Fermi level is e ressed in Eqn. 8-17 by the sum of... 

Base configs. Single exc. Double exc. Total State... 

The existence of the states H H+ and H+H is not in disagreement with the properties of the hydrogen molecule, e.g. the absence of a dipole moment. Since the states H+H and H H+ certainly have the same energy, they are equally frequent and the total state... 

All conservation laws together define a linear subspace of lattice points in the total state space. The accessible subspace lies in this subspace and usually is identical with it, but not necessarily so. A counterexample would be... 

The above procedures yield expressions for the states at a single point on the trough of minimum-energy points. As all of these points are equivalent, the total states of the system must be an integral... 

Thus the density of states, and total states, may be evaluated for a molecule with both vibration and internal rotation. 

This form is convenient for computation, requiring first the evaluation of J2(e) and W(E + ), and hence ft(E) over the energy range. W(t+) is, of course, the total states of the complex for a thermal energy of +, including vibration and internal rotation. 

The classical expression for the total states of s harmonic oscillators... 

Fig. 28.1. In a genetic switch, the total state of the system depends on two variables whether the DNA site is occupied or not and the number of copies of the transcription factor. In the left panel, the logarithm of the steady state probability for the occupancy state and protein number is plotted. This probability acts like an effective potential. In the right panel, the effective potential for a charge transfer or two site polaron is plotted as a function of the enviromnent polarization for the two electronic states. The governing time-dependent eqnations for the two problems share many similarities...

Py). (As we will see shortly, the and p will not play a role and have been dropped from the discussion.) In principle, total states can be generated from a single phase point. However, we must eliminate some states because certain combinations, such as F = (q,, q, p , p ) and F = (-q, -qy, -p, -Py), lead to identical nonequilibrium trajectories. If we further require that the shear stresses be equal in magnitude, we are left with the following four combinations ... 

List the liquids in order of decreasing rates of entropy. List the liquid that reached a total state of entropy most quickly first and so on. 

Reported PWS population exceeded total state population PWS population was set to the 2000 U.S. census of state populations. 

Quantitative results for high-energy processes may be obtained from a multiple-collision expansion. Indicating with k the free total state in channel k (a product of relative, internal and atomic spin states), the transition matrix element M31 for A + BC -> AB + C is given by... 

First proceed by considering the case s = 2. Recall that for a single oscillator, the density of states is (hv) and the total states for H is E Ihv. For two oscillators... 

---