Commutator Representation of Heisenberg equation hierarchy
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In this paper,by making use of spectral gradient mdthod, the operator pairs of Heisenberg equation is given and Heisenberg equation hierarchy is obtained. Then through resolving a key equation, the commutator representation of this hierachy of the Heisenberg equation is obtained.Keywords:
Heisenberg group
Commutator
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Heisenberg group
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Manifold (fluid mechanics)
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We develop an algebraic approach to studying the spectral properties of the stationary Schrödinger equation in one dimension based on its high-order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schrödinger operator by n×n matrices for any n∈N and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n×n matrices. The connection to so-called quasiexactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high-order Lie symmetries, corresponds to exactly solvable Schrödinger equations. A symmetry classification of Schrödinger equation admitting nontrivial high-order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schrödinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multidimensional linear and one-dimensional nonlinear Schrödinger equations are briefly discussed.
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We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
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Time evolution
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The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are expressed in terms of pseudo-bosons, which do not behave as ordinary bosons under the adjoint transformation, but obey the Weil-Heisenberg commutation relations.
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Self-adjoint operator
Spectral Analysis
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The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in three-dimensional space-time: the standard three-wave equations and less known modified three-wave equations. The quantized systems in discrete space-time may be understood as the regularized integrable quantum field theories. Integrability of the theories, and in particular the quantum tetrahedron equations for vertex operators, follow from the quantum auxiliary linear problems. Principal object of the lattice field theories is the Heisenberg discrete time evolution operator constructed with the help of vertex operators.
Geometric Quantization
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We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to $c$-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where $c$-number and $q$-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
Heisenberg picture
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Energy operator
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This brief article in the first section brings out the fact that the Heisenberg and the Schrodinger operators become identical if the operators correspond to a conserved quantity. In Section II solutions we show that for time independent operators 𝐴𝑆 that satisfy [𝐴𝑆,𝐻]=0 where 𝐻 satisfies the Schrodinger equation 𝑖ℏ𝜕𝜓𝜕𝑡=𝐻𝜓 we have 𝐴𝑆=𝑎 where ‘a’ is an eigen value of the stated equation. It is independent of space and time coordinates.
Schrödinger's cat
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Heisenberg picture
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