论文标题
量子谐波振荡器和经典振荡器的等效性受到随机力
Equivalence of quantum harmonic oscillators and classical oscillators subject to random forces
论文作者
论文摘要
我们表明，量子谐波振荡器的schrödinger方程可以作为与$ o（m / \ hbar）$的随机力的牛顿力学的近似，当时$ \ hbar / m \ hbar / m \ ll 1 $。相反，包括所有叠加状态在内的Schrödinger方程的每个解决方案都以这种方式出现。换句话说，量子谐波振荡器大约只有经典的谐波振荡器，其质量和频率与量子谐波振荡器相同，但要受到随机力。我们将结果推广到多个非相互作用的振荡器。 We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces.这对于时间间隔是有效的$ O（\ tilde {m} / \ hbar）$，其中$ \ tilde {m} $是$ \ hbar / \ tilde {m} {m} \ ll 1 $时，是最小质量振荡器的质量。相反，每个解决方案，包括所有纠缠状态，都以这种方式出现。换句话说，$ n $非相互作用的量子谐波振荡器大约只有$ n $非相互作用的经典谐波振荡器，其质量和频率与量子振荡器相同。这提供了当地的牛顿纠缠量子振荡器的纠缠模型。纠缠状态所需的相关性嵌入经典振荡器的相空间概率密度中。
We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals $O( \tilde{m} / \hbar)$, where $\tilde{m}$ is the mass of the minimum mass oscillator, when $\hbar/\tilde{m} \ll 1$. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, $n$ non-interacting quantum harmonic oscillators are approximately nothing but $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.
