View in EDS

A simple quantum dot: Numerical and variational solutions.

Saved in:
Bibliographic Details
Title: A simple quantum dot: Numerical and variational solutions.
Authors: Walsh, Connor M.1 (AUTHOR) cmwalsh@ualberta.ca, MacPherson, Ian1 (AUTHOR), Joseph, Davidson Noby1 (AUTHOR), Kabra, Suyash1 (AUTHOR), Toor, Ripanjeet Singh1 (AUTHOR), Protter, Mason1 (AUTHOR), Marsiglio, Frank1,2 (AUTHOR) fm3@ualberta.ca
Source: American Journal of Physics. Jun2026, Vol. 94 Issue 6, p465-475. 11p.
Subjects: Bound states, Finite differences, Physics education, Variational approach (Mathematics), Quantum dot devices, Matrix mechanics, Numerical analysis
Abstract: We describe a simple quantum dot that consists of two crossed two-dimensional troughs. As such there is no potential well; nonetheless, this geometry gives rise to a bound state, centered on the point at which these troughs cross one another. This problem is interesting both because the existence of a bound state may surprise students and because it can be solved using a variety of computational techniques, including matrix mechanics, finite differences, and mode matching. We present these methods and show how the mode-matching method in this case provides the most accurate solution to the problem. Additionally, the mode-matching method can be used to generate a simple wave function that yields the lowest energy known to date to arise out of an analytical variational solution for this problem. Editor's Note: The one-dimensional potential well is a standard exercise in quantum mechanics, familiar to every student as an introduction to bound states and quantization. This paper extends that paradigm to a 2D situation in which two perpendicular troughs cross, forming a simple quantum dot defined purely by geometry rather than by a conventional closed well. Classically, no bound state is expected; nevertheless, quantum mechanically, the intersection region localizes a quantum particle and supports a true bound state. This problem is also used to illustrate and put into practice several standard methods such as matrix mechanics, finite differences, and mode matching. Appropriate for advanced quantum mechanics classes as well as for courses dealing with numerical methods in this field. [ABSTRACT FROM AUTHOR]
Copyright of American Journal of Physics is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 193951313
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: A simple quantum dot: Numerical and variational solutions.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Walsh%2C+Connor+M%2E%22">Walsh, Connor M.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> cmwalsh@ualberta.ca</i><br /><searchLink fieldCode="AR" term="%22MacPherson%2C+Ian%22">MacPherson, Ian</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Joseph%2C+Davidson+Noby%22">Joseph, Davidson Noby</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kabra%2C+Suyash%22">Kabra, Suyash</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Toor%2C+Ripanjeet+Singh%22">Toor, Ripanjeet Singh</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Protter%2C+Mason%22">Protter, Mason</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Marsiglio%2C+Frank%22">Marsiglio, Frank</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> fm3@ualberta.ca</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22American+Journal+of+Physics%22">American Journal of Physics</searchLink>. Jun2026, Vol. 94 Issue 6, p465-475. 11p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Bound+states%22">Bound states</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+differences%22">Finite differences</searchLink><br /><searchLink fieldCode="DE" term="%22Physics+education%22">Physics education</searchLink><br /><searchLink fieldCode="DE" term="%22Variational+approach+%28Mathematics%29%22">Variational approach (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+dot+devices%22">Quantum dot devices</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+mechanics%22">Matrix mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We describe a simple quantum dot that consists of two crossed two-dimensional troughs. As such there is no potential well; nonetheless, this geometry gives rise to a bound state, centered on the point at which these troughs cross one another. This problem is interesting both because the existence of a bound state may surprise students and because it can be solved using a variety of computational techniques, including matrix mechanics, finite differences, and mode matching. We present these methods and show how the mode-matching method in this case provides the most accurate solution to the problem. Additionally, the mode-matching method can be used to generate a simple wave function that yields the lowest energy known to date to arise out of an analytical variational solution for this problem. Editor's Note: The one-dimensional potential well is a standard exercise in quantum mechanics, familiar to every student as an introduction to bound states and quantization. This paper extends that paradigm to a 2D situation in which two perpendicular troughs cross, forming a simple quantum dot defined purely by geometry rather than by a conventional closed well. Classically, no bound state is expected; nevertheless, quantum mechanically, the intersection region localizes a quantum particle and supports a true bound state. This problem is also used to illustrate and put into practice several standard methods such as matrix mechanics, finite differences, and mode matching. Appropriate for advanced quantum mechanics classes as well as for courses dealing with numerical methods in this field. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of American Journal of Physics is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=193951313
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1119/5.0275290
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 11
        StartPage: 465
    Subjects:
      – SubjectFull: Bound states
        Type: general
      – SubjectFull: Finite differences
        Type: general
      – SubjectFull: Physics education
        Type: general
      – SubjectFull: Variational approach (Mathematics)
        Type: general
      – SubjectFull: Quantum dot devices
        Type: general
      – SubjectFull: Matrix mechanics
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
    Titles:
      – TitleFull: A simple quantum dot: Numerical and variational solutions.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Walsh, Connor M.
      – PersonEntity:
          Name:
            NameFull: MacPherson, Ian
      – PersonEntity:
          Name:
            NameFull: Joseph, Davidson Noby
      – PersonEntity:
          Name:
            NameFull: Kabra, Suyash
      – PersonEntity:
          Name:
            NameFull: Toor, Ripanjeet Singh
      – PersonEntity:
          Name:
            NameFull: Protter, Mason
      – PersonEntity:
          Name:
            NameFull: Marsiglio, Frank
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 06
              Text: Jun2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 00029505
          Numbering:
            – Type: volume
              Value: 94
            – Type: issue
              Value: 6
          Titles:
            – TitleFull: American Journal of Physics
              Type: main
ResultId 1