Calculating Hermitian Forms: The Importance of Considering Singular Points. (10th September 2019)

• Record Type: Journal Article
• Authors: Bhowmick, Somnath; Hagebaum‐Reignier, Denis; Jeung, Gwang‐Hi
• Is Part Of: Bulletin of the Korean Chemical Society. Volume 41:Number 3(2020)
• Page Start: 254
• Page End: 260
• Publication Date: 2019-09-10
• Subjects: Singular points -- Harmonic oscillator -- Hydrogen atom -- Green's function -- Heaviside step function; Chemistry -- Periodicals; 540.5
• DOI: 10.1002/bkcs.11871
• Languages: English
• ISSNs: 0253-2964

Abstract

Abstract : 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 using H ^ 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. Abstract : Wrong and true solutions for the calculation of expectation values in two simple models (one‐dimensional harmonic oscillator and three‐dimensional hydrogen atom): singular points matter!