布朗粒子与物质波
Brownian Particles and Matter Waves
AI总结：
本文结合微流变学与德布罗意物质波观测进展，在不预设量子退相干论证的前提下，分析两类场景下布朗粒子展现波粒二象性的条件，得出马尔可夫扩散不等价于量子力学、强束缚下才可能满足相关能量要求的结论。
摘要：
鉴于微流变学在监测尺寸小至数纳米的布朗粒子随机运动方面取得的显著进展，同时德布罗意物质波已在尺寸相当的纳米级大分子上被实验观测到，我们在不预先采用量子退相干相关论证的前提下，研究布朗粒子是否能展现波粒二象性。首先，我们考察布朗粒子浸没在具有不随时间变化的扩散系数的无记忆粘性流体中的情况；若要求布朗粒子展现波粒二象性，会得出扩散系数必须与时间的倒数成正比这一站不住脚的结论，即扩散系数在早期会发散。这一发现与过往结论一致——量子力学不等价于马尔可夫扩散过程。接下来，我们考察布朗粒子被束缚在存在耗散与无耗散的简谐势阱中的情况。存在耗散情形下的福克-普朗克（Fokker-Plank）方程解，以及无耗散情形下的薛定谔（Schrödinger）方程解，都得出了相同的物理上可接受的结果：若布朗粒子要展现波粒二象性，其平均动能需要为量子简谐振子基态能量的一半。我们的一维计算表明，要实现这一点，束缚作用必须极强，布朗纳米粒子需要被嵌入刚度极高的固体中。
英文摘要：
In view of the remarkable progress in micro-rheology to monitor the random motion of Brownian particles with size as small as few nanometers, in association that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size; we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient; and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time; therefore, diverging at early times. This finding agrees with past conclusions--that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker-Plank equation for the case with dissipation, and of the Schrodinger equation for the case without dissipation lead to the same physically acceptable result-that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy needs to be half the ground-state energy of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian nanoparticle needs to be embedded in an extremely stiff solid.
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