非厄密量子力学
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摘要
上世纪二十年代，在总结大量实验和旧量子论的基础上，建立了反映微观粒子运动规律的量子力学。人们开始深入地研究物质的微观结构，从而物质的物理和化学性能极其变化规律被进一步地掌握。反映这些物质特性的重要指标是可观测物理量，它们在量子力学中用算符来表示。可观测物理量与实际应用相联系，被当做量子理论的基本要求和前提，而厄密算符能与所有这些量子理论基础自洽，所以长期以来量子力学中用厄密算符表示可观测物理量成为一种根深蒂固的观念。但事实上可观测物理量只是算符具有厄密性的必要条件，很多非厄密算符也可以对应正定的实数本征值。这些非厄密算符也对应一套自洽的量子理论，这些理论大大地扩展了厄密量子理论的应用范围。
本论文主要研究非厄密量子力学中的PT对称理论和赝厄密量子理论，归纳如下：
通过对赝厄密量子理论的研究，我们在一个各向异性平面谐振子中加入一个正比于ip1p2的虚数作用项，从而构建了一个两维具有置换对称性的PT赝厄密哈密顿量模型，并证明了此模型等价于Pais-Uhlenbeck振子，从而在此PT对称赝厄密系统与四阶微分振子模型之间建立联系。我们还发现了模型中存在的置换对称性自发破缺机制，这种机制在给出模型的实数能谱中起着关键作用，并保证其能谱不会出现正负能级置换现象。进一步发现，我们模型中的二维置换对称性对应于Pais-Uhlenbeck振子中两个不相等频率的等价性（不是大小，而是属性），并揭示了在Pais-Uhlenbeck振子中作为前提要求的非等频条件可以合理地被解释为这种等价性的自发破缺。
我们构建了一种适用于赝厄密系统的代数方法，为此引入了算符η+，定义了新的刃矢量态和刁矢量态，并重新定义互为η+赝厄密共轭的产生算符和湮灭算符。作为应用，构建了一个宇称（P）赝厄密哈密顿量，并进行了详细分析，而且使用这种代数方法得到了此哈密顿量系统的实数能谱。还通过特殊选择的V算符确定了相应的算符η+=P V。对于V算符的选择来说，一方面要保证P赝厄密哈密顿量同时具有PV赝厄密自伴随性质，另一方面，PV还要确保系统具有实数能谱和正定内积。另外，当此P赝厄密哈密顿量系统被拓展到空间坐标算符和动量算符都不对易的正则非对易空间时，得出了一阶非对易修正的能谱。尤其是发现，非对易性并没有改变系统能谱的实数性和正定内积。
利用代数方法重新讨论了两个非厄密的PT对称哈密顿量系统，这种代数方法源于求解赝厄密哈密顿量系统而非PT对称系统。相比于将一个非厄密哈密顿量转化为相应的厄密哈密顿的方法来说，代数方法的优点是不改变PT对称哈密顿量的Hilbert空间。我们引入算符V来替代C，从而给出PT对称系统的正定内积。算符V与PT对称量子力学中通常采用的C算符有相同的作用，但它可以直接利用哈密顿量来构建，应用更加方便。最后得到了两个非厄密PT对称系统的能谱，这些结果与原文章中的一致，并且，我们还求出了这两个非厄密PT对称系统的正定内积。
推导了在二维坐标空间含有两个独立小量的非厄密微扰公式。运算过程中，用到非厄密系统的正定内积，而不是普通厄密量子力学中的正定内积，比如η+赝厄密内积或PTV内积。得到的本征态函数和能谱的微扰公式分别精确到两个小量的一阶和二阶。我们通过在一维自由谐振子中加入一个正比于ixp的非厄密项，构建了一个非厄密PT对称哈密顿量，并得到了它的实数能谱和本征函数。进而，此模型被推广到动量和坐标都不对易的非对易空间，然后利用推导的非厄密微扰公式得到了推广后分别精确到两个小量一阶和二阶的本征函数和能谱，而且验证了推广后的系统本征函数依然具有PTV正定内积。从对易空间推广到非对易空间，非厄密PT对称哈密顿量系统的性质没有发生改变。
In the1920s, based on the conclusion of lots of experiments and the old quantumtheory, new quantum mechanics was built to describe the principal of motion of micro-scopic particles. The microstructure of the matter was started to be researched deeply,and then the physical and chemical properties of the matter as well as its principal ofmotion were revealed further. Observables that describe the properties of the matterplay a crucial role, they are represented by operators in quantum mechanics. The ob-servables associated with the application in reality, thus, the observability is regardedas the prerequisite and necessary condition in the quantum mechanics. The Hermitianoperator is consistent with all the requirements mentioned above. So we take it forgranted that the obervables are characterized by Hermitian operators for a long time.In fact, the observable is only a necessary condition of the Hermiticity of the opera-tor. Many non-Hermitian operators also have real solutions, and they comply with aset of self-consistent quantum theory different from the one satisfied by the Hermitianoperator. These new ideas greatly expand the application of the Hermitian quantummechanics.
This thesis mainly deals with the non-Hermitian P T symmetric theory and thepseudo-Hermitian quantum theory. Our work are organized as follows:
By adding an imaginary interacting term proportional to ip1p2to the Hamilto-nian of a free anisotropic planar oscillator, we construct a new model which isdescribed by the P T-pseudo-Hermitian Hamiltonian with the permutation sym-metry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our P T-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also pointout the spontaneous breaking of permutation symmetry which plays a crucialrole in giving a real spectrum free of interchange of positive and negative energylevels in our model. Moreover, we find that the permutation symmetry of twodimensions in our Hamiltonian corresponds to the identity (not in magnitude butin attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, andreveal that the unequal-frequency condition imposed as a prerequisite upon thePais-Uhlenbeck oscillator can reasonably be explained as the spontaneous break-ing of this identity.
An algebraic method for pseudo-Hermitian Hamiltonian systems is proposedthrough introducing the operator η+, defining new bra and ket vector states andredefining annihilation and creation operators to be η+-pseudo-Hermitian (notHermitian) adjoint of each other. As an example, a parity-pseudo-HermitianHamiltonian is constructed and analyzed in detail. Its real spectrum is obtainedby means of the algebraic method, where the corresponding operator η+is foundto be P V through a specific choice of V. The operator V is given in such a waythat on the one hand this P-pseudo-Hermitian Hamiltonian is also P V-pseudo-Hermitian self-adjoint and on the other hand P V ensures a real spectrum and apositive-definite inner product. Moreover, when the P-pseudo-Hermitian systemis extended to the canonical noncommutative space with noncommutative spatialcoordinates and noncommutative momenta as well, the first order noncommu-tative correction of energy levels is calculated, and in particular the reality ofenergy spectra and the positive-definiteness of inner products are found to be notaltered by the noncommutativity.
Two non-Hermitian P T-symmetric Hamiltonian systems are reconsidered bymeans of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the P T-symmetric ones. Com-pared with the way converting a non-Hermitian Hamiltonian to its Hermitiancounterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian P T-symmetric Hamiltonian unchanged. In order to give the positivedefinite inner product for the P T-symmetric systems, a new operator V, insteadof C, can be introduced. The operator V has the similar function to the operatorC adopted normally in the P T-symmetric quantum mechanics, however, it canbe constructed, as an advantage, directly in terms of Hamiltonians. The spectraof the two non-Hermitian P T-symmetric systems are obtained, which coincidewith that given in literature, and in particular, the Hilbert spaces associated withpositive definite inner products are worked out.
Non-Hermitian perturbation formulas with two independent small parameters intwo dimensions are deduced, where the usual positive inner products in Hermi-tian quantum mechanics are replaced by non-Hermtian positive inner products,such as the η+-pseudo-Hermitian inner products or the P T V inner products. Theformulas of eigenstates and the spectrum are given up to the first and secondorder of the two small parameters respectively. We propose a non-Hermitian but PT symmetric Hamiltonian by adding a non-Hermitian term proportional toixp to the free harmonic oscillator Hamiltonian, and obtain its real spectrum andeigenstates. Moreover, the model is generalized to noncommutative space withnoncommutative spatial coordinate operators and noncommutative momentumoperators. We get the eigenstates and real spectrum up to the first and second or-der of the two small parameters respectively in this case, and the eigenstates co-incide with the positive P TVinner products. The properties of the non-HermtianP T symmetric system keep unchange when it is generalized from the usual spaceto the noncommutative one.
本论文主要研究非厄密量子力学中的PT对称理论和赝厄密量子理论，归纳如下：
通过对赝厄密量子理论的研究，我们在一个各向异性平面谐振子中加入一个正比于ip1p2的虚数作用项，从而构建了一个两维具有置换对称性的PT赝厄密哈密顿量模型，并证明了此模型等价于Pais-Uhlenbeck振子，从而在此PT对称赝厄密系统与四阶微分振子模型之间建立联系。我们还发现了模型中存在的置换对称性自发破缺机制，这种机制在给出模型的实数能谱中起着关键作用，并保证其能谱不会出现正负能级置换现象。进一步发现，我们模型中的二维置换对称性对应于Pais-Uhlenbeck振子中两个不相等频率的等价性（不是大小，而是属性），并揭示了在Pais-Uhlenbeck振子中作为前提要求的非等频条件可以合理地被解释为这种等价性的自发破缺。
我们构建了一种适用于赝厄密系统的代数方法，为此引入了算符η+，定义了新的刃矢量态和刁矢量态，并重新定义互为η+赝厄密共轭的产生算符和湮灭算符。作为应用，构建了一个宇称（P）赝厄密哈密顿量，并进行了详细分析，而且使用这种代数方法得到了此哈密顿量系统的实数能谱。还通过特殊选择的V算符确定了相应的算符η+=P V。对于V算符的选择来说，一方面要保证P赝厄密哈密顿量同时具有PV赝厄密自伴随性质，另一方面，PV还要确保系统具有实数能谱和正定内积。另外，当此P赝厄密哈密顿量系统被拓展到空间坐标算符和动量算符都不对易的正则非对易空间时，得出了一阶非对易修正的能谱。尤其是发现，非对易性并没有改变系统能谱的实数性和正定内积。
利用代数方法重新讨论了两个非厄密的PT对称哈密顿量系统，这种代数方法源于求解赝厄密哈密顿量系统而非PT对称系统。相比于将一个非厄密哈密顿量转化为相应的厄密哈密顿的方法来说，代数方法的优点是不改变PT对称哈密顿量的Hilbert空间。我们引入算符V来替代C，从而给出PT对称系统的正定内积。算符V与PT对称量子力学中通常采用的C算符有相同的作用，但它可以直接利用哈密顿量来构建，应用更加方便。最后得到了两个非厄密PT对称系统的能谱，这些结果与原文章中的一致，并且，我们还求出了这两个非厄密PT对称系统的正定内积。
推导了在二维坐标空间含有两个独立小量的非厄密微扰公式。运算过程中，用到非厄密系统的正定内积，而不是普通厄密量子力学中的正定内积，比如η+赝厄密内积或PTV内积。得到的本征态函数和能谱的微扰公式分别精确到两个小量的一阶和二阶。我们通过在一维自由谐振子中加入一个正比于ixp的非厄密项，构建了一个非厄密PT对称哈密顿量，并得到了它的实数能谱和本征函数。进而，此模型被推广到动量和坐标都不对易的非对易空间，然后利用推导的非厄密微扰公式得到了推广后分别精确到两个小量一阶和二阶的本征函数和能谱，而且验证了推广后的系统本征函数依然具有PTV正定内积。从对易空间推广到非对易空间，非厄密PT对称哈密顿量系统的性质没有发生改变。
In the1920s, based on the conclusion of lots of experiments and the old quantumtheory, new quantum mechanics was built to describe the principal of motion of micro-scopic particles. The microstructure of the matter was started to be researched deeply,and then the physical and chemical properties of the matter as well as its principal ofmotion were revealed further. Observables that describe the properties of the matterplay a crucial role, they are represented by operators in quantum mechanics. The ob-servables associated with the application in reality, thus, the observability is regardedas the prerequisite and necessary condition in the quantum mechanics. The Hermitianoperator is consistent with all the requirements mentioned above. So we take it forgranted that the obervables are characterized by Hermitian operators for a long time.In fact, the observable is only a necessary condition of the Hermiticity of the opera-tor. Many non-Hermitian operators also have real solutions, and they comply with aset of self-consistent quantum theory different from the one satisfied by the Hermitianoperator. These new ideas greatly expand the application of the Hermitian quantummechanics.
This thesis mainly deals with the non-Hermitian P T symmetric theory and thepseudo-Hermitian quantum theory. Our work are organized as follows:
By adding an imaginary interacting term proportional to ip1p2to the Hamilto-nian of a free anisotropic planar oscillator, we construct a new model which isdescribed by the P T-pseudo-Hermitian Hamiltonian with the permutation sym-metry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our P T-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also pointout the spontaneous breaking of permutation symmetry which plays a crucialrole in giving a real spectrum free of interchange of positive and negative energylevels in our model. Moreover, we find that the permutation symmetry of twodimensions in our Hamiltonian corresponds to the identity (not in magnitude butin attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, andreveal that the unequal-frequency condition imposed as a prerequisite upon thePais-Uhlenbeck oscillator can reasonably be explained as the spontaneous break-ing of this identity.
An algebraic method for pseudo-Hermitian Hamiltonian systems is proposedthrough introducing the operator η+, defining new bra and ket vector states andredefining annihilation and creation operators to be η+-pseudo-Hermitian (notHermitian) adjoint of each other. As an example, a parity-pseudo-HermitianHamiltonian is constructed and analyzed in detail. Its real spectrum is obtainedby means of the algebraic method, where the corresponding operator η+is foundto be P V through a specific choice of V. The operator V is given in such a waythat on the one hand this P-pseudo-Hermitian Hamiltonian is also P V-pseudo-Hermitian self-adjoint and on the other hand P V ensures a real spectrum and apositive-definite inner product. Moreover, when the P-pseudo-Hermitian systemis extended to the canonical noncommutative space with noncommutative spatialcoordinates and noncommutative momenta as well, the first order noncommu-tative correction of energy levels is calculated, and in particular the reality ofenergy spectra and the positive-definiteness of inner products are found to be notaltered by the noncommutativity.
Two non-Hermitian P T-symmetric Hamiltonian systems are reconsidered bymeans of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the P T-symmetric ones. Com-pared with the way converting a non-Hermitian Hamiltonian to its Hermitiancounterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian P T-symmetric Hamiltonian unchanged. In order to give the positivedefinite inner product for the P T-symmetric systems, a new operator V, insteadof C, can be introduced. The operator V has the similar function to the operatorC adopted normally in the P T-symmetric quantum mechanics, however, it canbe constructed, as an advantage, directly in terms of Hamiltonians. The spectraof the two non-Hermitian P T-symmetric systems are obtained, which coincidewith that given in literature, and in particular, the Hilbert spaces associated withpositive definite inner products are worked out.
Non-Hermitian perturbation formulas with two independent small parameters intwo dimensions are deduced, where the usual positive inner products in Hermi-tian quantum mechanics are replaced by non-Hermtian positive inner products,such as the η+-pseudo-Hermitian inner products or the P T V inner products. Theformulas of eigenstates and the spectrum are given up to the first and secondorder of the two small parameters respectively. We propose a non-Hermitian but PT symmetric Hamiltonian by adding a non-Hermitian term proportional toixp to the free harmonic oscillator Hamiltonian, and obtain its real spectrum andeigenstates. Moreover, the model is generalized to noncommutative space withnoncommutative spatial coordinate operators and noncommutative momentumoperators. We get the eigenstates and real spectrum up to the first and second or-der of the two small parameters respectively in this case, and the eigenstates co-incide with the positive P TVinner products. The properties of the non-HermtianP T symmetric system keep unchange when it is generalized from the usual spaceto the noncommutative one.
引文
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