{"title":"与三对角线哈密顿量相关的相干态","authors":"Hashim A. Yamani, Zouhaïr Mouayn","doi":"10.1016/S0034-4877(23)00059-9","DOIUrl":null,"url":null,"abstract":"<div><p><span>It has been shown that a positive semi-definite Hamiltonian </span><em>H</em><span>, that has a tridiagonal matrix representation in a given basis\n</span><span><math><mrow><mrow><mo>{</mo><mrow><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow></mrow><mo>}</mo></mrow></mrow></math></span>, can be represented in the form <em>H</em> = <em>A<sup>†</sup>A</em>, where <em>A</em> is a forward shift operator satisfying\n<span><math><mrow><mi>A</mi><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow><mo>=</mo><msub><mi>c</mi><mi>n</mi></msub><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow><mo>+</mo><msub><mi>d</mi><mi>n</mi></msub><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub><mo>-</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math></span> playing the role of an annihilation operator. Here, the coherent states |<em>z</em><span>) are defined as eigenstates of </span><em>A</em>. We also exhibit a complete set of coherent states {|<em>c<sub>n</sub></em>)}, labeled by the discrete points <em>c<sub>n</sub></em>, which we may also use as a basis. We find a solution of the coherent state |<em>z</em>), indexed by the continuous parameter <em>z</em> as a series expansion in terms of the {|<em>c<sub>n</sub></em>)}. We further show how to compute the time development of the coherent state |<em>z</em><span>) and we illustrate this with examples. As a major result, we find the explicit closed form solution |</span><em>z</em>) for the Morse oscillator.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Coherent states associated with tridiagonal Hamiltonians\",\"authors\":\"Hashim A. Yamani, Zouhaïr Mouayn\",\"doi\":\"10.1016/S0034-4877(23)00059-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>It has been shown that a positive semi-definite Hamiltonian </span><em>H</em><span>, that has a tridiagonal matrix representation in a given basis\\n</span><span><math><mrow><mrow><mo>{</mo><mrow><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow></mrow><mo>}</mo></mrow></mrow></math></span>, can be represented in the form <em>H</em> = <em>A<sup>†</sup>A</em>, where <em>A</em> is a forward shift operator satisfying\\n<span><math><mrow><mi>A</mi><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow><mo>=</mo><msub><mi>c</mi><mi>n</mi></msub><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub></mrow><mo>〉</mo></mrow><mo>+</mo><msub><mi>d</mi><mi>n</mi></msub><mrow><mo>|</mo><mrow><msub><mi>ϕ</mi><mi>n</mi></msub><mo>-</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math></span> playing the role of an annihilation operator. Here, the coherent states |<em>z</em><span>) are defined as eigenstates of </span><em>A</em>. We also exhibit a complete set of coherent states {|<em>c<sub>n</sub></em>)}, labeled by the discrete points <em>c<sub>n</sub></em>, which we may also use as a basis. We find a solution of the coherent state |<em>z</em>), indexed by the continuous parameter <em>z</em> as a series expansion in terms of the {|<em>c<sub>n</sub></em>)}. We further show how to compute the time development of the coherent state |<em>z</em><span>) and we illustrate this with examples. As a major result, we find the explicit closed form solution |</span><em>z</em>) for the Morse oscillator.</p></div>\",\"PeriodicalId\":49630,\"journal\":{\"name\":\"Reports on Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Reports on Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0034487723000599\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487723000599","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
证明了在给定基{| n >}中具有三对角矩阵表示的正半定哈密顿量H可以表示为H = a†a,其中a是满足a | n > =cn| n > +dn| n-1 >的正移算子,起湮灭算子的作用。在这里,相干态|z)被定义为a的特征态。我们还展示了一个完整的相干态集合{|cn)},由离散点cn标记,我们也可以将其用作基。我们找到了相干态|z)的解,它由连续参数z表示为{|cn)}的级数展开。我们进一步展示了如何计算相干态的时间发展(z),并举例说明了这一点。作为主要结果,我们找到了莫尔斯振子的显式闭形式解(z)。
Coherent states associated with tridiagonal Hamiltonians
It has been shown that a positive semi-definite Hamiltonian H, that has a tridiagonal matrix representation in a given basis
, can be represented in the form H = A†A, where A is a forward shift operator satisfying
playing the role of an annihilation operator. Here, the coherent states |z) are defined as eigenstates of A. We also exhibit a complete set of coherent states {|cn)}, labeled by the discrete points cn, which we may also use as a basis. We find a solution of the coherent state |z), indexed by the continuous parameter z as a series expansion in terms of the {|cn)}. We further show how to compute the time development of the coherent state |z) and we illustrate this with examples. As a major result, we find the explicit closed form solution |z) for the Morse oscillator.
期刊介绍:
Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.