The innovation engine for new materials

Anna Barth

Anna Barth




Harvey Mudd College


Vojtěch Vlček,

Faculty Sponsor(s): 

Vojtěch Vlček,

Faculty Sponsor's Department(s): 

Chemistry and Biochemistry

Project Title: 

Toy Models for Testing the Efficiency of Stochastic Sampling in Hilbert Space

Project Description: 

An understanding of materials’ electronic properties requires calculations that involve excited
states of electrons. These calculations can be completed in linear time by a stochastic
approach, which involves sampling random vectors in the electrons’ Hilbert space. However, in
practice, it has been observed that the stochastic fluctuations are higher for strongly localized
electronic states than for other states, which increases the computational cost in those cases.
In order to gain a better understanding of why the stochastic fluctuations may be higher for
certain states, we present several toy models that exhibit similar behavior and can be more
easily analyzed than many-body electronic systems . Specifically, we perform simple calculations
on energy eigenstates of a modified quantum harmonic oscillator, using a Python
implementation of the stochastic approach. In these toy models, we find that strongly localized
states exhibit lower stochastic fluctuations. However, by selecting a finely spaced representation
in position space, the stochastic fluctuations can be made uniform across all states. Our
analysis provides a simple case study of how the representation of quantum states can lead to
an uneven distribution of fluctuations in stochastic calculations.