Using the symbol calculus developed by Berezin, Kree and Raczka, it is shown that the algebra generated by the canonical commutation relation (CCR) is splitted into creation part and annhilation part with holomorphic ...Using the symbol calculus developed by Berezin, Kree and Raczka, it is shown that the algebra generated by the canonical commutation relation (CCR) is splitted into creation part and annhilation part with holomorphic and anti-bolomorphic symbols respectively. Further. in the Weyl representation of CCR, three Weyl operators will constitute an irreducible set of the fall CCR-algebra if some number theoretic conditions are satisfied.展开更多
We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation t...We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.展开更多
Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl tran...Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl transformation can immediately lead to Eresnel transformation kernel in classical optics.展开更多
By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weyl ordering expansion of power product of coordinate and momentum operators （√2Q）^m（√2iP） ^τ=：： Hm,r （√2...By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weyl ordering expansion of power product of coordinate and momentum operators （√2Q）^m（√2iP） ^τ=：： Hm,r （√2Q, √2iP）：：, the introduction of two-variable Hermite polynomial Hm,r brings much convenience to the study of Weyl correspondence.展开更多
We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl orderedproduct of operators,i.e.,the Weyl correspondence rule can be reconstructed by classical functions' Fourier t...We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl orderedproduct of operators,i.e.,the Weyl correspondence rule can be reconstructed by classical functions' Fourier transfor-mation followed by an inverse Fourier transformation within Weyl ordering of operators.As an application of thisreconstruction,we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.展开更多
Using the Weyl ordering of operators expansion formula （Hong-Yi Fan, J. Phys.A 25 （1992） 3443） this paper finds a kind of two-fold integration transformation about the Wigner operator △（ q＇,p） q-number transf...Using the Weyl ordering of operators expansion formula （Hong-Yi Fan, J. Phys.A 25 （1992） 3443） this paper finds a kind of two-fold integration transformation about the Wigner operator △（ q＇,p） q-number transform） in phase space quantum mechanics,∫∫∞-∞dp＇dq＇/π △（q＇,p＇）e-2i（ p-p＇）（ q-q＇）=δ（ p-P）δ（ q-Q）,∫∫∞-∞dqdpδ（p-P）δ（q-Q）e2i（p-p＇）（q-q＇）=△（q＇,p＇）,whereQ,P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q-P ordering, P-Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.展开更多
In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊...In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊-A operator, then Weyl＇s theorem holds for f（T）, where f is an analytic functions on some neighborhood of σ（T） and not constant on each of the components of its domain.展开更多
Using the correspondence between psedodifferential operator and its symbol,the authors obtain Heisenberg's inequality in Sobolev spaces and therefore a kind of quantitatire representation of uncertainty principle.
文摘Using the symbol calculus developed by Berezin, Kree and Raczka, it is shown that the algebra generated by the canonical commutation relation (CCR) is splitted into creation part and annhilation part with holomorphic and anti-bolomorphic symbols respectively. Further. in the Weyl representation of CCR, three Weyl operators will constitute an irreducible set of the fall CCR-algebra if some number theoretic conditions are satisfied.
基金National Natural Science Foundation of China under Grant Nos.10775097 and 10874174
文摘We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.
基金Supported by the National Natural Science Foundation of China under Grant No.10475056the Research Foundation of the Education Department of Jiangxi Province
文摘Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl transformation can immediately lead to Eresnel transformation kernel in classical optics.
基金Supported by the President Foundation of Chinese Academy of Scienceby the Specialized Research Fund for the Doctorial Progress of Higher Education of China
文摘By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weyl ordering expansion of power product of coordinate and momentum operators （√2Q）^m（√2iP） ^τ=：： Hm,r （√2Q, √2iP）：：, the introduction of two-variable Hermite polynomial Hm,r brings much convenience to the study of Weyl correspondence.
基金supported by the Specialized Research Fund for the Doctorial Progress of the Higher Education of China under Grant No.20040358019the National Natural Science Foundation of China under Grant No.10775097
文摘We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl orderedproduct of operators,i.e.,the Weyl correspondence rule can be reconstructed by classical functions' Fourier transfor-mation followed by an inverse Fourier transformation within Weyl ordering of operators.As an application of thisreconstruction,we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10775097 and 10874174)Specialized Research Fund for the Doctoral Program of Higher Education of China
文摘Using the Weyl ordering of operators expansion formula （Hong-Yi Fan, J. Phys.A 25 （1992） 3443） this paper finds a kind of two-fold integration transformation about the Wigner operator △（ q＇,p） q-number transform） in phase space quantum mechanics,∫∫∞-∞dp＇dq＇/π △（q＇,p＇）e-2i（ p-p＇）（ q-q＇）=δ（ p-P）δ（ q-Q）,∫∫∞-∞dqdpδ（p-P）δ（q-Q）e2i（p-p＇）（q-q＇）=△（q＇,p＇）,whereQ,P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q-P ordering, P-Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.
基金partially supported by the NNSF(11201126)the Basic Science and Technological Frontier Project of Henan Province(142300410167)+1 种基金the Natural Science Foundation of the Department of Education,Henan Province(14B110008)the Youth Science Foundation of Henan Normal University(2013QK01)
文摘In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊-A operator, then Weyl＇s theorem holds for f（T）, where f is an analytic functions on some neighborhood of σ（T） and not constant on each of the components of its domain.
文摘Using the correspondence between psedodifferential operator and its symbol,the authors obtain Heisenberg's inequality in Sobolev spaces and therefore a kind of quantitatire representation of uncertainty principle.
