Scaled Schrodinger equation

Eqs. (5) and (7) can be thought of as changes in the distance and energy units, respectively. In these new, dimension-scaled, units the radial expectation value, (r) = has a much tamer dimension dependence, and the new groimd-state energy, E = — is completely independent of D. Thus, the one-dimension limit of the scaled Schrodinger equation, Eq. (6), is not such a bad model for the physical, three-dimensional, problem. 

Although the H2 problem for D-dimensions is separable in spheroidal coordinates, just as for D = 3, since we want to examine the nonseparable situation, we employ cylindrical coordinates. In these coordinates the nuclei are located on the z-axis at —i /2 and -t-iJ/2, respectively, and the electron is at p,z). Dimensional scaling is introduced by using units of jZ bohr radii for distance and hartrees for energy, with Z the nuclear charge and k = D — l)/2. The scaled Schrodinger equation for H then takes a simple form. 

Scaling procedure, 219, 269 Schmidt procedure, 283 Schrodinger equation for an electronic system, 212... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

Strictly speaking, a chiral species cannot correspond to a true stationary state of the time-dependent Schrodinger equation H

time scale for such spontaneous racemization is extremely long. The wavefunction of practical interest to the (finite-lived) laboratory chemist is the non-stationary Born-Oppenheimer model Eq. (1.2), rather than the true T of Eq. (1.1). 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

The overall scheme of ab initio molecular dynamics is similar to that of classical molecular dynamics described earlier but instead of using interatomic potentials, the Schrodinger equation is solved to provide the energy and the forces acting on the particles. The computational cost is huge and most studies are limited to small simulation cells (< 100 atoms) and time-scales of a few picoseconds. Within... 

D. A. Mazziotti, Linear scaling and the 1,2-contracted Schrodinger equation. J. Chem. Phys. 115, 8305 (2001). 

In order to use wave-function-based methods to converge to the true solution of the Schrodinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both FIF and post-HF calculations. Even with these improvements, however you can appreciate the problem with... 

With the scaling in Eq. (8), the one-particle hydrogen-like Schrodinger equation without external fields transforms as... 

J Bengtsson, E. Lindroth, S. Selsta, Solution of the time-dependent Schrodinger equation using uniform complex scaling, Phys. Rev. A 78 (3) (2008) 032502. 

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