Visualization of Wave Functions

Eamonn F. Healy

Ki Soo Chung

Alan C. Calder, Michael Curry, Robert M. Panoff +1 more

An essential part of variational Monte Carlo or Green's function Monte Carlo (GFMC) algorithms is the trial wave function. In the case of particles obeying Fermi statistics, this wave function is antisymmetric and cannot be interpreted directly as a probability distribution, thereby making calculations difficult. Some progress can be made in GFMC algorithms, however, by requiring the trial wave function to have the same ``fixed'' nodes as the variational function. The sensitivity of the energy to changes in the nodal surface has remained a significant unresolved issue for two reasons: (a) the many-dimensional nodal surface is hard to quantify or visualize and (b) it has been difficult to vary this surface in a controlled fashion. As a first step toward gaining further information, we have developed a method of quantifying the many-dimensional nodal surface by creating and visually characterizing a related surface. The related surface is the surface of the local volume in which any given particle can range without changing the overall sign of the wave function, and this surface is mapped for different configurations and wave functions. A comparison of these surfaces allows us to characterize the local contributions to the nodal surface of a particular wave function and thereby visually differentiate between different types of wave functions. We show that introducing backflow correlations into the trial wave function results in a constrained volume with the undesired effect of significantly slowing down GFMC calculations. This study demonstrates that advanced visualization methods can serve a useful role in the process of algorithm development rather than just in the presentation of results. \textcopyright{} 1996 The American Physical Society.

David Austen, Wytze Brouwer, Bailey Martin

Cecilia Yanuarief

Research has been conducted which shows the quantization visualization of hydrogen atom orbitals angular space in Rosen Morse potential system interference when the electrons are excited in the state of nr= 2 and l = 1 through the polar function analysis of the Schrödinger Potential of Non-Central Coloumbic Rosen Morse. The polar Schrödinger equation is solved using the Romanovski polynomial method to obtain the polar wave function. To show the accuracy of the analysis, the polar wave function spectrum i s visualized with Matlab-based computer programming.

Scott C. Johnson, T. D. Gutierrez

A phonon often is described as “a quantum of lattice vibration,” but this description can be difficult to reconcile with the wave functions explored in a typical undergraduate quantum mechanics class. A phonon wave function is similar to the harmonic oscillator wave functions studied in introductory quantum mechanics, except that it is many-dimensional. We suggest a way to visualize the probability density for this very high-dimensional wave function. The resulting pictures are especially clear and intuitive for a coherent state, which is both a good approximation to a sound wave and a discrete analog to laser light. These pictures can also provide a qualitative introduction to quantum field theory.