BOX 2.1 The Schrödinger Wave Equation

The time evolution of a (nonrelativistic) quantum mechanical system is prescribed by the Schrödinger wave equation. For a particle with mass m and position r that is moving under the influence of a potential V(r), this equation reads:

where H is the linear Hermitian Hamiltonian operator:

Here is Planck's constant divided by 2π and has the very small value 1.05 x 10-27 erg-second characteristic of the submicroscopic quantum regime. The wavefunction ψ generally is complex; its amplitude squared ¦ψ¦2 provides the probability distribution for the position of the particle at time t. The linearity of the Schrödinger wave equation and the Hermiticity of H guarantees conservation of probability.

Time-independent solutions to the wave equation that have physical significance fall into two categories and require two distinct boundary conditions. The first (''bound states'') are square-integrable eigenfunctions of H; they are bounded, vanish at infinity, and provide a discrete spectrum of real energy eigenvalues for H. The second ("scattering states") are a continuum of solutions that lie in energy above the infinite-r limit of V(r) (if it exists), and the wavefunction remains nonzero (and oscillatory as a plane wave or sinusoidal function) in this asymptotic limit.

Quantum mechanical phenomena implied by the Schrödinger wave equation have a strongly counterintuitive flavor. The discreteness of bound-state energies contrasts starkly with the continuous energies available to a dynamical system in classical mechanics. The corresponding discreteness of the energy spectrum of transitions between pairs of quantum bound states has spawned the colloquially overused and misused phrase "quantum leap." In addition, the wavefunction can be nonvanishing in regions where V exceeds the total energy, leading to the well-known, but often mystifying, tunneling phenomenon; this permits particles to pass through potential energy barriers that in classical mechanics would provide absolute blockage. Less well known, but equally mystifying, is the reverse phenomenon originally pointed out by von Neumann and Wigner (1929): some potential functions V(r) have the capacity to trap particles in square-integrable eigenstates at energies above the absolute maximum of V, again in contradiction to classical mechanical behavior.

The Schrödinger wave equation adapts naturally to the description of many mutually interacting particles. The case most prominently considered in chemistry is that of two or more electrons, where V includes Coulombic interactions for electron-electron repulsions and electron-nucleus attractions. However, it is important to account for the fact that electrons in particular are endowed with intrinsic angular momentum, denoted by "spin," that is quantized to display only two allowed values, "up" and "down." In the strictly nonrelativistic regime (chemically, that which involves only atoms with low atomic numbers), these electron spins are invariants of motion and can be formally eliminated from the mathematical problem, provided that the remaining spatial wavefunction for the electrons satisfies appropriate symmetry conditions. In the case of two-electron systems (examples are the helium atom and the hydrogen molecule), the spatial wavefunction must either be symmetric under interchange of electron positions (if one "up" spin and one "down" spin are present) or antisymmetric (if both spins are "up" or both are "down"). Analogous, but more complicated, interchange conditions are applicable to the spatial wavefunction when more electrons are present.

Reference

von Neumann, J., and E. Wigner, 1929, Phys. Z. 30:465.



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