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arXiv 2102.09534gr-qcquant-ph
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引力潮汐力对自由下落量子粒子影响的新观察

Fresh look at the effects of gravitational tidal forces on a freely-falling quantum particle

F. Hammad, P. Sadeghi, N. Fleury, A. Leblanc

摘要:

本文对球对称引力场中的量子粒子自由降落过程中潮汐力的影响作了进一步的研究。我们从完全相对论性的克莱因-戈登方程出发,导出相应的粒子的薛定谔方程,以便(i)简要讨论等效原理的问题,(ii)能够比较方程中的相对论项与潮汐力项。在二阶非相对论近似下,所得到的薛定谔方程在水平方向上是简谐振子的方程,在垂直方向上是反谐振子的方程。两种方法用于在垂直方向上求解方程。第一种方法是基于一个固定的边界条件,并产生一个离散的能量谱的波函数是渐近的一个粒子在一个线性引力场。第二种方法是基于随时间变化的边界条件,并产生随时间衰减的量化能量谱。移动到一个自由下落的参考系,我们得到相应的随时间变化的能谱。潮汐力的作用产生了哈密顿量的期望值和波包宽度随时间的相对变化,它们与质量无关。等效原理,我们在这里理解为引力和惯性之间的经验等效,讨论这些不同的结果的基础上。为了完整起见,我们简要地讨论了预期将获得的结果为玻色-爱因斯坦凝聚体或超流体在自由降落使用非线性Gross-Pitaevskii方程。

英文摘要:

We take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schrödinger equation for the particle by starting from the fully relativistic Klein-Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schrödinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepacket's width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose-Einstein condensate or a superfluid in free fall using the nonlinear Gross-Pitaevskii equation.

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