Posmom : The Unobserved Observable

We have recently shown that the probability density S(s) of the position-momentum dot product s=r 3p of a particle can be computed efficiently from its wave function ψ(r). Here, by examining the H atom and LiH molecule, we show that S(s) yields insight into the nature of electronic trajectories, and we argue that electron posmometry provides information that is inaccessible by diffraction or momentum methods. SECTION Molecular Structure, Quantum Chemistry, General Theory O neof the first successful attempts to understand electronic behavior in matter was Bohr's famous model of the hydrogen atom. By quantizing the angular momentumwith an integern (the principal quantumnumber), he sought to describe electronic motion using circular orbits, analogous to those of a planet. Some years later, Sommerfeld refined this model, quantizing the z component of the angular momentum with another integer l to yield the Old QuantumTheory. In thismodel, the electron follows elliptical orbits. Although the Bohr-Sommerfeld (BS)model appeared to explain a number of features of atomic spectra, its insistence that electrons follow classical trajectories is essentially incorrect and has been superseded by modern quantum mechanical treatments. The advent of the Heisenberg Uncertainty Principle, which states that the position r and the momentum p of a particle cannot be known simultaneously with arbitrary precision, showed that the classical concept of a trajectory begins to lose its significance in nanoscopic systems and must be abandoned completely in discussing the motion of an electron. In such circumstances, where classical mechanics aims to predict the position r of the particle at a time t, quantummechanics more modestly (and correctly) offers only a wave function ψ(r)which is related to the probability of finding the particle at the position r at time t. In nonrelativistic wave mechanics, the wave functions ψn(r) and energies En of a particle are found by solving the Schr€ odinger equation Hψn = Enψn where H is the Hamiltonian operator for the system. As Dirac discovered, these position wave functions are related to the particle's momentum wave functions φn(p) by a Fourier transform, 7 and this is depicted as a horizontal orange arrow in Figure 1. The fact that the product of the variance of a function and the variance of its Fourier transform is strictly positive is then the Schr€ odinger explanation for the Uncertainty Principle. In the Born interpretation, the squared modulus of the wave function yields the corresponding probability density and, in this way, one can form the position density F(r)=|ψ(r)| andmomentum densityΠ(p)=|φ(p)| of the particle (Figure 1, blue arrows). Unlike the wave functions, the densities are experimental observables, and it is possible to measure F(r) by X-ray or neutron diffraction and to measure Π(p) by Compton spectroscopy. Such techniques are widely used for the characterization of matter in condensed phases and furnish valuable information about molecular structure and bonding. We note that, although ψ(r) and φ(p) contain the same information, the density F(r) contains information that is not present in Π(p), and vice versa. In this way, position and momentum spectroscopy provide complementary perspectives. The Fourier transforms in atomic units (the orange arrows at the top and bottom of Figure 1) of the position and momentum densities F̂ðpÞ 1⁄4 Z FðrÞeir 3 p dr ð1Þ Π̂ðrÞ 1⁄4 Z ΠðpÞeir 3 p dp ð2Þ are also important, and the convolution theorem shows that Π̂(r) is the autocorrelation of the position wave function Π̂ðrÞ 1⁄4 Z ψ/ðr0 þ r=2Þψðr0 r=2Þ dr0 ð3Þ and F̂(p) is the autocorrelation of the momentum wave function F̂ðpÞ 1⁄4 Z φ/ðp0 þ p=2Þφðp0 -p=2Þ dp0 ð4Þ These connections are depicted by purple arrows in Figure 1. Thus, to compute the momentum density from the position wave function, one can take either the squared modulus of the Fourier transform, or the Fourier transform of the autocorrelation. Received Date: February 16, 2010 Accepted Date: March 18, 2010