arXiv 1306.58032013-06-24math-phmath.MPquant-ph

含二阶导数的拉氏量的Ostrogradsky哈密顿量的Feynman路径积分

Feynman's Path Integral to Ostrogradsky's Hamiltonian for Lagrangians with second derivatives

G. E. Hahne

备注：: 13 pages, 0 figures

摘要：

计算表明，费曼的路径积分意味着Ostrogradsky的哈密顿量的非奇异拉格朗日二阶导数。该方法采用稳相近似法求出了单位时间内波函数的极限变化。通过介绍，该方法重新适用于只有一阶导数的非奇异拉格朗日量的情况下，但不一定是二次的速度。计算的副产品是将一般经典拉格朗日函数转换为哈密顿函数的勒让德变换的替代推导。在一阶和二阶导数的情况下，结果精确地包含经典的哈密顿量，它代表了在某个时刻作用在波函数上的伪微分算子（不一定是厄米算子）的所谓“符号”。本文的推导论证了一个主张，即费曼的方法开始于一个经典的拉格朗日和结束于一个经典的哈密顿-非经典的运营商订购规定的通道，从经典的量子哈密顿需要外部输入，一般不是固有的费曼的路径积分形式主义。

英文摘要：

A calculation is presented that shows that Feynman's path integral implies Ostrogradsky's Hamiltonian for nonsingular Lagrangians with second derivatives. The procedure employs the stationary phase approximation to obtain the limiting change of the wave function per unit time. By way of introduction, the method is applied anew to the case of nonsingular Lagrangians with only first derivatives, but not necessarily quadratic in the velocities. A byproduct of the calculation is an alternate derivation of the Legendre transformation of taking general classical Lagrangians into Hamiltonians. In both the first and second derivative cases, the outcome contains precisely the classical Hamiltonian, which represents the so-called "symbol" of a (not necessarily Hermitean) pseudodifferential operator acting on the wave function at an instant of time. The derivation herein argues for a claim that Feynman's method starts with a classical Lagrangian and ends with a classical Hamiltonian---nonclassical operator-ordering prescriptions in the passage from classical to quantum Hamiltonians require external input, and are generally not inherent in Feynman's path integral formalism.

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