Degenerate stationary states

Figure 8.9 shows the various degenerate stationary-state boundaries from Fig. 6.18 as broken curves, and also (solid curve) a locus corresponding to Po k2 values for which the system can have a stationary-state solution with two eigenvalues simultaneously and exactly equal to zero. Mathematically the conditions for this are... 

States of the system may be divided into three groups (a) nonstationary state, VxF 0 (b) nondegenerate stationary state, VxF=0, det(F,) 0 (Vij = d2V/dxidxJ) (c) degenerate stationary state Vx F = 0, det(fy) = 0 analogously to the classification used in elementary catastrophe theory. 

Stationary states of a gradient system may be said to lie on the catastrophe surface M and degenerate stationary states to be placed on the singularity set I (see Chapter 2). 

Exact properties of such a state have not been found. Guckenheimer studied the properties of the system nearby a less degenerate stationary state in which the polynomial W(k) has one zero solution for k > 1, JF(l) has a pair of purely imaginary roots while the remaining roots of the equations W(n) have negative real parts. Then, the control parameters satisfy the requirements... 

Let us consider two non-degenerate stationary states m and n of a system, with energy values Wm and Wn such that Wm is greater than Wn. According to the Bohr frequency rule, transition from one state to another will be accompanied by the emission or absorption of radiation of frequency... 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

Different combinations of degenerate orbitals simplify different problems. For example, the normal representations px, py, pz of the / = 1 orbitals are actually mixtures of different // / values. The stationary states with a single value of mi 0 have real and imaginary parts, and are difficult to visualize. 

In a hydrogen atom, the 2x and 2p orbitals are degenerate. Therefore, as discussed in the last section, any other combination of these orbitals would also be a stationary state with the same energy. The only reason for writing these orbitals as we did in Equa-... 

In the formation of hybrid orbitals it is assumed that the proximity of other atoms so modifies the character of the one 2s and the three 2p orbitals of an atom that their energies are sufficiently close to make them degenerate (see Stationary States, Chapter 6). Then any linear combination of them is also a possible orbital. In particular, you can make the combinations... 

These are the two mutually-degenerate stationary ground states of the noninteracting H atoms a and b describes the state in which electron 1 is around proton a and electron 2 is around proton b. In 2. the electrons are exchanged. When the two H atoms interact at a proton spacing R, the degeneracy is lifted (see Eqns. (Pl.l) and (Pl-2)). The symbols used there are the overlap integral. 

As an example, the stationary-state wave functions 211. 121, and I m for the particle in a cubic box are degenerate, and the linear combination Ci 2ii + 2) 121 + is an eigenfunction of the particle-in-d-cubic-box Hamiltonian with eigenvalue the same eigenvalue as for each of 2in i2i> and 2. 

At high vibrational excitation, for each value of V there is always a lowest energy group of N nearly degenerate levels, corresponding to V quanta of excitation in one bond and none in the other N-1 bonds, represented by the local mode wavefunctions [V,0,0,. .. ]. The stationary state... 

Fig. 5.3. A degenerate state may mean an instable system "superpolarizfjble ), which adapts itself very easily to an external (even very small) perturbation. The figure shows how the hydrogen atom in a degenerate state changes when a proton approaches it. (a) the atom in a non-perturbed stationary state 2px (b) The proton interacts with the hydrt en atom, but the 2px orbital does not reflect this. It is true that the orbital 2px describes the total energy vecy well (because the interaction is weak), but it describes the electron charge distribution in the hydrogen atom interacting with the proton very badly, (c) Contrary to this, the function = (2s) + + 2py), satisfies the Schrddinger equation fo" the unperturbed hydrt en atom as well as the function...

The stationary-state perturbation theory used in this section is applicable only to nondegenerate states degenerate perturbation theory must... 

Radiationless transitions occur even in dilute gases indicating that such processes are possible in isolated molecules. If a molecule were excited to a stationary state of the Hamiltonian the matrix element V would, of necessity, be zero. However, excitation at a precise energy is not possible unless exceptionally monochromatic radiation is used (as could be the case with laser excitation). Hence, the energetic distinction between nearly degenerate vibronic states is lost if the line width of the exciting radiation is sufficiently large. As a result transition may be to a nonstationary state. See A. H. Zewail, T. E. Orlowski, and K. E. Jones, Proc. Nat. Acad. Sci, U.S.A. 74, 1310 (1977). 

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