Eigenfunctions degenerate

For a degenerate system we cannot always determine beforehand the zero-order or model functions. The mixing of the degenerate eigenfunctions in the zero-order depends on the perturbation. Therefore, generally speaking, we can only confine the model function to a certain subspace of the entire functional space. Let us call this subspace the model (or P)... 

Equation (2) shows that the band energy has the symmetry of the point group of the wave vector. As R runs over the whole R P(k), it generates a set of degenerate eigenfunctions... 

If ra is an irreducible representation of dimension k, and if F, F2, is a set of degenerate eigenfunctions that form the basis for theyth irreducible representation of the group of symmetry operations, these eigenfunctions transform according to the relation... 

The set of spatial functions d>K, which may be either exact degenerate eigenfunctions of the Hamiltonian operator H or approximations to them, must then provide the associate representation D in which (using ep for the parity factor, 1 for P even or odd)... 

The lowest-energy solutions deviating from spherieal symmetry are the 2p-orbitals. Using Eqs (7.44), (7.45) and the f = 1 spherical harmonics, we find three degenerate eigenfunctions ... 

A contour plot is shown in Fig. 7.8. Note that this function is cylindrically-symmetrical about the z-axis with a node in the x, y-plane. The eigenfunctions 21 1 are complex and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics 7i i, shown in Fig. 6.4. As deduced in Section 4.2, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine,... 

It is easy to prove that a linear combination of degenerate eigenfunctions is itself an eigenfunction of the same energy. Let... 

It can be proved that the eigenvalues of hermitian operators are real. Accordingly, observables are represented by hermitian operators, as that guarantees that the outcome of observations is real. Another property of those operators is that their non-degenerate eigenfunctions are orthogonal, that is... 

It is clear from Eq. (2.69) that equal values of energy can be obtained using different sets of quantum numbers Ux, riy and n, that is, degenerate eigenfunctions of the hamiltonian exist. Figure 2.5 shows plots of two... 

The theorem just mentioned can also be applied to the 2s and 2p functions which are degenerate eigenfunctions of the hamiltonian. Therefore, any linear combination of them is also an eigenfunction of H. For example. 

Then g(l>j is still an eigenfunction of H with eigenvalue e. But the ,),=].are linearly independent and a linear combination of degenerate eigenfunctions is still an eigenfunction with the same eigenvalue, then we must have... 

Conserve the overlaps between orthogonal degenerate eigenfunctions. 

Let ,)o,.. . , 0 )o be arbitrarily chosen degenerate eigenfunctions of h with true counterparts ,) =/c (/),)o. Applying an arbitrary linear transformation to the produces other arbitrary model eigenfunctions i ) 0 = S I ) 0 with overlaps... 

In equation (36), y denotes a set of due-degenerate eigenfunctions of the one-electron Hamiltonian h, while TV denotes the number of states in this set. Taking the Hermitian conjugate of equation (36) gives us the corresponding pair annihilation... 

A and / are considered as the degenerate eigenfunctions of the absorbing molecule in the absence of a magnetic field. The functions that describe the states A and / and their corresponding zeroth-order energies are... 

We assume that, when the stationary non-degenerate eigenfunction... 

Prove the following statement any linear combination of degenerate eigenfunctions of H is also an eigenfunction of H. 

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