Hamiltonian operator variable corresponding

From this expression one has to remove the center-of-mass motion. The corresponding Hamiltonian can then be rewritten in terms of the intrinsic variables and the Euler angles. The general form of the Hamiltonian operator is, in the Eckart frame,1... 

One of the postulates of quantum-mechanical theory is that for every mechanical variable there is a corresponding mathematical operator. For example, the operator that corresponds to the mechanical energy is the Hamiltonian operator, and the time-independent SchrOdinger equation is the eigenvalue equation for this operator. For motion in the X direction of a single particle of mass m with a potential energy given by V(x), the Hamiltonian operator is... 

We need to find the operator that corresponds to a particular mechanical variable. We begin by assuming that the Hamiltonian operator is the mathematical operator that is in one-to-one correspondence with the energy of a system. This is plausible since the time-independent Schrodinger equation... 

The Hamiltonian operator in Eq. (25.1-1) corresponds to a time-independent Schrodinger equation that can be solved by separation of variables, using a trial solution ... 

By replacing variables in the classical Hamiltonian with corresponding quantum mechanical operators, develop the quantum mechanical Hamiltonian operator and express the Schrodinger equation specific to the problem. 

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

When this operator is substituted into equation (7.246), it corresponds to a small rotation in rotational-spin-orbital space. The parameter. si governs the magnitude of this rotation. It is a variable parameter which can be chosen to eliminate terms from the transformed Hamiltonian. Using this form for F and the well known commutation relations between the molecule-fixed components of J, S and L, it is easy to show that... 

Because H is dynamically dependent on spin and space variables, the expression in parentheses in the r.h.s. of Eq. (3) involving integration over the latter defines a spin operator. This is just the effective Hamiltonian of interest to us. By virtue of point (iii), when the integrations are to be performed for the H" term in the Hamiltonian, only the unit operator in A need to be retained. The resulting expression will thus have the form (Ap H"l ). If one takes into account that the space state 1 ) is a product (or a combination of products, see above) of localized, one-particle states, one can immediately see that upon integrating over the spatial variables r , n= 1,2,...,AI, the spatial parts of the individual spin-dependent terms will be replaced by the corresponding quantum mechanical averages. Thus, for the entire expression in Eq. (3) is none other than one of the matrix element of the standard NMR Hamiltonian, Wnmr, between two spin-product basis states,... 

The Schrodinger equation corresponding to a given classical Hamiltonian is then obtained by replacing all of the dynamical variables in the original Hamiltonian with their operator analogs. 

If the last term in (28) is ignored, then what is left seems to be the right sort of nuclear motion Hamiltonian for present purposes. The function Ep(q) + F"(q) is clearly a potential in the nuclear variables and the kinetic energy operator depends only on the nuclear variables too. It is not quite true that (25) is actually the clamped nuclei Hamiltonian but it is shown in (7) that the correspondence is very close. 

Having the Hamiltonian treatment, it is easy to proceed to a quantum mechanical formulation of the Fermi resonance problem. The classical variables (9.15) and their complex conjugates correspond to the annihilation and creation operators of a quantum oscillator ... 

The derivation of the Hamiltonian resembles the standard procedure the classical Lagrange function is constructed first, then it is used to express the classical Hamilton function and then quantisation is applied by substituting the canonical variables for corresponding quantum-mechanical operators. There are two additional requirements the Hamiltonian should be symmetric with respect to the interchange of two electrons, and it should be Hermitian. 

Let Ai, A2, , Ale, he a, group of symmetry operators that commute with the Hamiltonian 3C. The subjection of the variables of a function iff to the point symmetry transformation At,. - , will be written as TAt f, where is the linear operator. The group requirements are that = TAtJU, where Ta ai corresponds to some other... 

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