Hamiltonian energy eigenvalues

Since the Hamiltonian operator is hermitian, the energy eigenvalues E are real. 

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set ms, mr), corresponding to the electron and nuclear spin quantum numbers ms and mr, respectively. The energy eigenvalues of eqn (1.7) are ... 

From (27) and (29) it follows that every component of the total angular momentum operator J = L + S and J2 commute with the Dirac Hamiltonian. The eigenvalues of J2 and Jz are j(j + 1 )h2 and rrijh respectively and they can be defined simultaneously with the energy eigenvalues E. 

A better method is the average t-matrix approximation (ATA) (Korringa 1958), in which the alloy is characterized by an effective medium, which is determined by a non-Hermitean (or effective ) Hamiltonian with complex-energy eigenvalues. The corresponding self-energy is calculated (non-self-... 

From the density of a given excited state one can obtain the Hamiltonian, the eigenvalues and eigenfunctions, and (through adiabatic connection) the noninteract-ing effective potential V" °. The solution of equations of the noninteracting system then leads to the density nt. Thus, we can consider the total energy to be a functional of the noninteracting effective potential ... 

This Hamiltonian is identical to that of stretching vibration [Eq. (6.7)]. The only difference is that the coefficients A, in front of C, are related to the parameters of the potential, D and a, in a way that is different for Morse and Poschl-Teller potentials. The energy eigenvalues of uncoupled Poschl-Teller oscillators are, however, still given by... 

The quantum version of the Hamiltonian Eq. (7) has been studied for decades in both Physics and Chemistry in the 2-level limit. If the potential energy surface (PES) is represented as a quartic double well, then the energy eigenvalues are doublets separated by, roughly, the well frequency. When the mass of the transferred particle is small (e g. electron), or the barrier is very high, or the temperature is low, then only the lowest doublet is occupied this is the 2-level limit of the Zwanzig Hamiltonian. 

Note that as long as the unitary transformation is exact, i.e., as long as the Taylor series is infinite and converges, the exact energy eigenvalues of the 4-component untransformed Hamiltonian are obtained. 

Figure 13.7 Example of potential energy curves for separate VB Hamiltonians and the curve for the lowest energy eigenvalue when the separate Hamiltonians are coupled by an off-diagonal term in a 2x2 Hamiltonian matrix. Note that the difference between the minima for and 7/22 is the term r2 in the example given in the text only if all non-bonded and electrostatic interactions are identical in the two VB representations, and thus the quotes around the label...

One finds that states of a given electronic configuration that have the same values of L and 5 have the same energy eigenvalue for the Hamiltonian H°+ H such states are said to belong to the same term. Different terms... 

Show that addition of a constant C to a time-independent Hamiltonian H leaves the stationary-state wave functions unchanged and adds C to each energy eigenvalue. 

Figure 5.8 Energy eigenvalues (dots) of the discretized and complex rotated one-particle Dirac Hamiltonian with s-symmetry. Bound states that are well represented in the box and on the grid are lying on the x-axis. The low-energy pseudo-continuum states are rotated with an angle of approximately 20 (where 0 is the complex rotation angle) down from the real axis, as indicated with the straight solid line.

In the recent work [BAL85] we give a detailed study of various level schemes for odd-odd nuclei obtained by analytical solutions of Hamiltonians associated with either (2) or (3). The isomorphisms between the two group chains are elaborated for the case where the unpaired nucleons occupy same or all of levels with j = 1/2,3/2,5/2 and the analytical expressions for the corresponding energy eigenvalues are given. 

In the case of a scattering resonance, bound-free correlation is modified by a transient bound state of fV+1 electrons. In a finite matrix representation, the projected (fV+l)-electron Hamiltonian H has positive energy eigenvalues, which define possible scattering resonances if they interact sufficiently weakly with the scattering continuum. In resonance theory [270], this transient discrete state is multiplied by an energy-dependent coefficient whose magnitude is determined by that of the channel orbital in the resonant channel. Thus the normalization of the channel orbital establishes the absolute amplitude of the transient discrete state, and arbitrary normalization of the channel orbital cannot lead to an inconsistency. 

In work on the electronic structure of solids, Lowdin[40] pointed out that if the Hamiltonian matrix for a system were a polynomial function of the overlap matrix of the basis, H and S would have the same eigenvectors and the energy eigenvalues would be polynomial functions of the eigenvalues of S. A number of consequences of this sort of relationship are known, but so far as the author is aware, no tests of such an idea have ever been made with realistic H and S matrices. This may be accomplished by examining the commutator, since if... 

The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator ... 

Consider a molecule interacting with a pulse of coherent light, where the light is described by a purely classical field of Eq. (1.35) and the molecule is treated quantum mechanically. The dynamics of the radiation-free molecule is fully described by the (discrete or continuous) set of energy eigenvalues and eigenfunctions, denoted, respectively, as En and E ), of the material Hamiltonian Hu [Eq. (1-43)],... 

To find the stationary states ( eigenfunctions ) and the energies ( eigenvalues ) of the Hamiltonian we need to solve the Schrodinger equation ... 

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