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Calculating Hermitian Forms: The Importance of Considering Singular Points
Author(s) -
Bhowmick Somnath,
HagebaumReignier Denis,
Jeung GwangHi
Publication year - 2020
Publication title -
bulletin of the korean chemical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.237
H-Index - 59
ISSN - 1229-5949
DOI - 10.1002/bkcs.11871
Subject(s) - heaviside step function , classification of discontinuities , mathematics , operator (biology) , singularity , upper and lower bounds , hermitian matrix , function (biology) , harmonic oscillator , gaussian , mathematical analysis , quantum mechanics , pure mathematics , physics , biochemistry , chemistry , repressor , evolutionary biology , biology , transcription factor , gene
In this paper, we point out the importance of boundary conditions in the evaluation of expectation values of quantum mechanical operators involved in upper bound ( i . e ., variational principle) or lower bound ( e . g ., methods usingH ^ 2 ) calculations. The existence of singular points (or discontinuities) either in the trial function or in the operator itself needs to be carefully handled when calculating integrals, otherwise leads to non‐physical ( e . g ., imaginary) expectation values or to false values. In this case, the use of generalized functions ( e . g ., Heaviside or Dirac functions) is necessary to cure the singularity problems. As examples to put a stress on this mathematical subtleties, we discuss the wrong and true solutions obtained for the calculation of the mean values of the H ^ and T ^ V ^ (and V ^ T ^ ) operators using two standard simple models: the one‐dimensional harmonic oscillator and the three‐dimensional hydrogen atom, along with Slater‐type and Gaussian trial functions.