Hamiltonian operator eigenfunctions

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

Thus, the spatial function (q) is actually a set of eigenfunctions t/ n(q) of the Hamiltonian operator H with eigenvalues E . The time-independent Schrodin-ger equation takes the form... 

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

Given the Hamiltonian eqn (3.1), it is reasonable to express the eigenfunctions in terms of the electron and nuclear spin quantum numbers ms,mi). Applying to this function only the two terms in the Hamiltonian operator that involve the -direction of the field B we get ... 

In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

The first of these equations is called the time-independent Schrodinger equation it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. 

Because the Hamiltonian operator defined by Eq. (4.32) is separable, its many-electron eigenfunctions can be constructed as products of one-electron eigenfunctions. That is... 

In addition, the availability of HF wave functions made possible the testing of how useful such wave functions might be for the prediction of properties other than die energy. Simply because the HF wave function may be arbitrarily far from being an eigenfunction of the Hamiltonian operator does not a priori preclude it from being reasonably close to an eigenfunction for some other quantum mechanical operator. 

In magnetic resonance we are often confronted with the problem of obtaining a solution to a Hamiltonian which has only spin operators. To find the allowed energies and eigenfunctions, we generally start out with a convenient set of spin functions < , which represent the spin system but are not eigenfunctions of the Hamiltonian,. The eigenfunction ifi can, however, be constructed from a linear sum of the s ... 

Here,>/ is the Hamiltonian operator, which indicates that certain operations are to be carried out on a function written to its right. The wave equation states that, if the function is an eigenfunction, the result of performing the operations indicated by J( will yield the function itself multiplied by a constant that is called an eigenvalue. Eigenfunctions are conventionally denoted P, and the eigenvalue, which is the energy of the system, is denoted E. 

Here x) stands for the positional coordinates of all the particles in the system, E is the energy of the system, and 77 is the Hamiltonian operator. Since a symmetry operator merely rearranges indistinguishable particles so as to leave the system in an indistinguishable configuration, the Hamiltonian is invariant under any spatial symmetry operator R. Let tpi denote a set of eigenfunctions of H so that... 

Here if is the eigenfunction of the Hamiltonian operator H and the corresponding eigenvalue E is the energy of the system. 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

Table 3.1 Eigenfunctions Harmonic Oscillator H = — i//Jx) of the Hamiltonian Operator for the h2/2m) d2/dx2) + knX2...

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