Degeneracies and the Stark effect
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص445-446
2025-12-04
79
Degeneracies and the Stark effect
The energy of a symmetric rotor depends on J and K, and each level except those with K=0 is doubly degenerate: the states with K and −K have the same energy. However, we must not forget that the angular momentum of the molecule has a component on an external, laboratory-fixed axis. This component is quantized, and its permitted values are MJh, with MJ = 0, ±1,..., ±J, giving 2J + 1 values in all (Fig. 13.14). The quantum number MJ does not appear in the expression for the energy, but it is necessary for a complete specification of the state of the rotor. Consequently, all 2J + 1 orientations of the rotating molecule have the same energy. It follows that a symmetric rotor level is 2(2J + 1)-fold degenerate for K ≠ 0 and (2J + 1)-fold degenerate for K = 0. A linear rotor has K fixed at 0, but the angular momentum may still have 2J +1 components on the laboratory axis, so its degeneracy is 2J + 1. A spherical rotor can be regarded as a version of a symmetric rotor in which A is equal to B: The quantum number K may still take any one of 2J + 1 values, but the energy is independent of which value it takes. Therefore, as well as having a (2J + 1) fold degeneracy arising from its orientation in space, the rotor also has a (2J + 1)-fold degeneracy arising from its orientation with respect to an arbitrary axis in the therefore (2J + 1)2. This degeneracy increases very rapidly: when J = 10, for instance, there are 441 states of the same energy. The degeneracy associated with the quantum number MJ (the orientation of the rotation in space) is partly removed when an electric field is applied to a polar molecule (e.g., HCl or NH3), as illustrated in Fig. 13.15. The splitting of states by an electric field is called the Stark effect. For a linear rotor in an electric field E, the energy of the state with quantum numbers J and M-J is given by
E(J,MJ) = hcBJ(J + 1) + a(J,MJ)µ2E2
where (see Further reading for a derivation)
Note that the energy of a state with quantum number MJ depends on the square of the permanent electric dipole moment, µ. The observation of the Stark effect can therefore be used to measure this property, but the technique is limited to molecules that are sufficiently volatile to be studied by rotational spectroscopy. However, as spectra can be recorded for samples at pressures of only about 1 Pa and special techniques (such as using an intense laser beam or an electrical discharge) can be used to vapor ize even some quite nonvolatile substances, a wide variety of samples may be studied. Sodium chloride, for example, can be studied as diatomic NaCl molecules at high temperatures.
Fig. 13.14 The significance of the quantum number MJ. (a) When MJ is close to its maximum value, J, most of the molecular rotation is around the laboratory z-axis. (b) An intermediate value of MJ. (c) When MJ=0 the molecule has no angular momentum about the z-axis All three diagrams correspond to a state with K = 0; there are corresponding diagrams for different values of K, in which the angular momentum makes a different angle to the molecule’s principal axis.
Fig. 13.15 The effect of an electric field on the energy levels of a polar linear rotor. All levels are doubly degenerate except that with MJ =0.
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