Schrodinger Hamiltonian

The nonrelativistic, electronic Schrodinger Hamiltonian operator, designated as H, is represented by... 

J. Aguilar, J.M. Combes, A class of analytic perturbations for one-body Schrodinger Hamiltonians, Commun. Math. Phys. 22 (1971) 269. 

What are these "integrals" to which we have referred From the fact that the Schrodinger Hamiltonian contains only one- and two-electron operators, it is straightforward to show [ 17] that most of the matrix elements [43] which arise in computing the SCF energy and its derivatives with respect to nuclear motion can be written in terms of integrals of the general form... 

Making no notational distinction between electrons and nuclei, the laboratory-fixed form of the Schrodinger Hamiltonian describing the molecule as a system of N particles is... 

Abstract When considering the work of Carl Ballhausen on vibrational spectra, it is suggested that his use of the Born-Oppenheimer approximation is capable of some refinement and extension in the light of later developments. A consideration of the potential energy surface in the context of a full Coulomb Schrodinger Hamiltonian in which translational and rotational motions are explicitly considered would seem to require a reformulation of the Born-Oppenheimer approach. The resulting potential surface for vibrational motion should be treated, allowing for the rotational motion and the nuclear permutational symmetry of the molecule. 

When comparing this U-equation for statistical operator with the corresponding one within Schrodinger picture, one can immediately identify that the role of the Schrodinger Hamiltonian is played formally by ... 

The correspondence principle states that the Schrodinger Hamiltonian may be derived through a substitution of p and r in the total energy function H of Hamiltonian mechanics by operator expressions p and r. Thus, the form of H is known in terms of p and r. 

Of course, the explicit form of the velocity operator depends on H. For the Schrodinger Hamiltonian, Eq. (4.46), we obtain the Schrbdinger velocity operator... 

Finally, we consider in this section the force law in quantum mechanics (Ehrenfest, 1927). From Heisenberg s equation of motion employing the Schrodinger Hamiltonian we have... 

For the Hamiltonian, we choose again the Schrodinger Hamiltonian, but now for an N-particle system of Eq. (4.48), in which all kinetic energy operators of the individual particles are simply summed as in classical physics. All constant and potential energy operators in V r, ri,..., tn) in the Hamiltonian commute with the density operator pr and therefore cancel. In nonrelativistic... 

---