Degenerate eigenvalues

The probability of measuring A = where is the eigenvalue associated with the normalized eigenfiinction ( ), is precisely equal to r> - For degenerate eigenvalues, the probability... 

For the hermitian matrix in review exereise 3b show that the pair of degenerate eigenvalues ean be made to have orthonormal eigenfunetions. 

Next, find the pair of eigenveetors assoeiated with the degenerate eigenvalue of -2. First, root one eigenveetor one ... 

Notiee that the eigenvalue )ik = 3 appears twiee we say that the eigenvalue )ik = 3 is doubly degenerate. )ik = 2 is a non-degenerate eigenvalue. 

Each of these states is also an eigenstate of J2 because J2 and and, therefore, 2 and (e/+)n, commute. Furthermore, each of these states belongs to the single eigenvalue b2 of 2. The various (X + i Y)n>fi are degenerate eigenstates of J 2. 

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

For k = J and for degenerate eigenvalues the elements of f are taken equal to zero. Let us next consider the variation of the population-bond-order matrix, which, in the orthogonal case of eqn 1, is just ... 

There are now two possibilities the eigenvalue a, of A is either nondegenerate or degenerate. If a, is non-degenerate, then it corresponds to only one independent eigenfunction xpi, so that the function Bipi is proportional to... 

Thus, the function C, is an eigenfunction of both A and B with eigenvalues a, and Pi, respeetively. If a, and are non-degenerate, then there is only one eigenfunetion tpi eorresponding to them and the function Ctpi is proportional... 

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