简单非线性系统的半量子与半经典力学
Semiquantum versus semiclassical mechanics for simple nonlinear systems
摘要：
量子力学是在相空间中用维格纳函数作为量子密度算符的代表来表述的，而经典力学是在希尔伯特空间中用格罗内沃尔德算符作为经典刘维尔密度函数的代表来表述的。维格纳函数的量子演化的半经典近似已经被定义，使得量子演化可以从经典的出发点接近。现在类似的半量子近似的经典演化的Groenewold运营商的定义，使经典的演化接近从量子的起点。考虑了单自由度的简单非线性系统，其哈密顿量是简谐振子哈密顿量的多项式。的行为的期望值的简单的观测值和本征值的Groenewold运营商，数值计算和比较各种半经典和半量子近似。
英文摘要：
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behaviour of expectation values of simple observables and of eigenvalues of the Groenewold operator, are calculated numerically and compared for the various semiclassical and semiquantum approximations.
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