Operator domains and self-adjoint operators

Araujo, Vanilse S.; Coutinho, Francisco Antônio Bezerra; Perez, J. Fernando

doi:10.1119/1.1624111

Cited by 70 publications

(82 citation statements)

References 12 publications

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“…So we present below a result that helps to recognize if an operator is selfadjoint in its domain. This result is explained in reference [5], and is a basic criterion of self-adjointness. To see if an operator O is self-adjoint consider the equations…”

Section: Example 12) a Criterion Of Self-adjointnessmentioning

confidence: 99%

“…As shown in reference [5] (see also the Appendix 2), this is the most general boundary condition confining a particle to the right side of the real line that makes the Hamiltonian self-adjoint. The physical significance of the boundary conditions is as follows:…”

Section: Example 13) the Operator Momentummentioning

confidence: 99%

“…(Many other examples-which however are more advanced-of how to modify simple problems by specifying boundary conditions and their physical significance can be found in reference [5].) Example 1.4) A free particle confined to the right half of the real axis with the Hamiltonian defined in a domain where it is self-adjoint.…”

Section: Example 13) the Operator Momentummentioning

confidence: 99%

“…The distinction between Hermitian operators and self-adjoint operators is rather subtle. To clarify the distinction reference [5] presents two examples of operators with the same action but with different domains so that in the first example the operator is Hermitian and in the second the operator is self-adjoint. (One should note that the distinction between Hermitian and self-adjoint operators disappears for bounded operators-which are defined for all the functions of the Hilbert space-and hence for operators in finite dimensional spaces, that is for matrices.…”

Section: Self-adjointness and The Time-independent Schrödinger Eqmentioning

confidence: 99%

“…The objective of this paper is to make this link intuitively clear Until quite recently, domains or self-adjointness were not mentioned in the physical literature. However, in the last few years or so, some articles [4], [5], [6], [7] in the physics pedagogical literature begun to point out examples where domains of operators are essential to the full solution of the problems posed. We are aware of only three pedagogical articles published before those articles [6][7][8][9] that mention domains and self-adjointness.…”

Section: Introductionmentioning

confidence: 99%

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The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

2008

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We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems -the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone´s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the timedependent Schrödinger equation.

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Section: Example 12) a Criterion Of Self-adjointnessmentioning

confidence: 99%

Section: Example 13) the Operator Momentummentioning

confidence: 99%

Section: Example 13) the Operator Momentummentioning

confidence: 99%

Section: Self-adjointness and The Time-independent Schrödinger Eqmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

2008

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Calculating Hermitian Forms: The Importance of Considering Singular Points

Bhowmick

Hagebaum‐Reignier

Jeung

2019

Bulletin Korean Chem Soc

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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 〈〉trueH^bold2) 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 trueH^ and trueT^trueV^ (and trueV^trueT^) 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.

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Operator domains and self-adjoint operators

Cited by 70 publications

References 12 publications

The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

Calculating Hermitian Forms: The Importance of Considering Singular Points

Symbolic Computations

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