We give a partial answer to the question whether the Schrödinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a random force can be described by the Schrödinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson’s stochastic formulation of quantum mechanics for a random force depending on a small parameter. We show that the same result applies to small potential perturbations around the harmonic oscillator. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables. The main motivation of this paper is to provide a step for constructing a Newtonian theory which would approximately reproduce quantum mechanics both in unitary evolution and measurement regimes.

我们给出了一个问题的部分答案，即薛定ding方程是否可以从处于随机作用力的势能中的粒子的牛顿力学中导出。我们表明，一维谐波振荡器在受到随机力作用下的经典运动周围的波动可以通过薛定er方程描述一段时间，具体取决于振荡器的频率和能量。我们通过推导纳尔逊的随机力学量子公式的假设来实现这一点，其中随机力取决于一个小的参数。我们表明，相同的结果适用于谐波振荡器周围的小电势扰动。我们还表明，对于固定质量，可以选择噪声频谱以获得所有振荡器频率的结果。我们讨论启发式方法，以概括粒子在某一维度上的结果，其中可以使用动作角度变量来描述运动。本文的主要动机是为构建牛顿理论提供一个步骤，该理论将近似地重现单位演化和测量体系中的量子力学。