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Download A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson PDF

By E. Poisson

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Example text

Is limited to half of the j-values in either case. Insofar as the sum can be approximated by the integral method of (6) it has half the value given by (6), namely that of eq. (10). The case of Hg, for which the classical limit is inapplicable below room temperature, is discussed in more detail in the next section. 5. 9°K, so that even considerably above its low boiling point of 20°K no rotational levels are appreciably excited at equilibrium. However, there is an indirect effect of nuclear spin on the energy, which results in a curious complication.

W e discuss this in more detail in the next section. 2. 3) is a second order differential equation which must be solved subject to two different sets of conditions in order to find the states of the system. One of these sets of conditions is the set of boundary conditions, for instance that ψ vanish at the walls of the system. These conditions lead to a set Í l | of quantum numbers of states ψ^^^, with energies JS^^. The other conditions, the symmetry conditions, select from these functions, a limited number of linear combinations which are allowed.

For a molecule consisting of two isotopes, H D , H T , or D T , there is no symmetry requirement, since permutation of the two different nucleii not only permutes protons and neutrons but also displaces an extra neutron. All rotational states exist with each of the (2^1 - f 1) (2^2 + 1) different nuclear spin states. In the molecule Dg each deuterium nucleus has spin unity so that there are three states for each nucleus, ag, ag, or in the general case 2 ^ + 1 states, αχ, . , (x^s-v The 2 5 + 1 combinations a^{i) oi^(j) are necessarily symmetric, whereas from the remaining (25 + 1)^ — ( 2 « + 1) products av(i)a^(j) with ν Φ μ one can make l[{2s + l)^ — {2s + I)] symmetric and as many imsymmetric spin states, or in all ^(2s + 2){2s + 1) symmetric spin states, i(2s(2s + 1) antisymmetric spin states.

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