弹道粒子在拐点处经典禁区内的动量分布

Q3 Multidisciplinary
A. Villanueva
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引用次数: 0

摘要

单个粒子的波包𝚿𝚿(𝒙𝒙,)在其位置x和动量p之间具有统计相关性,量化为位置-动量协方差。协方差影响波包的传播和通过某一点的概率电流。本文给出了协方差的另一个效应:非零协方差可以表现为区域动量密度的不对称性。考虑一个选择性测量|𝚿𝚿⟩→|𝚿𝚿[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩在初始状态|𝚿𝚿⟩投射到状态|𝚿𝚿[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩较小的区域内𝒙𝒙𝟏𝟏<𝒙𝒙<𝒙𝒙𝟐𝟐。预计的动量表示状态是𝚽𝚽[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]=⟨𝒑𝒑|𝚿𝚿[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩和相应的动量密度|𝚽𝚽[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]|𝟐𝟐地区动量密度。本文研究了均匀重力作用下分子大小的粒子(用高斯波包𝚼𝚼表示)在经典拐点处的运动。本文考虑了𝚼𝚼的协方差对经典禁区区域动量密度的影响。具有给定协方差的初始状态𝚼𝚼皮肤病被投射到经典禁止区域,产生状态𝚼𝚼𝑪𝑪𝑪𝑪。如果相应的动力波函数是𝚽𝚽𝑪𝑪𝑪𝑪,区域动量密度是𝚷𝚷𝑪𝑪𝑪𝑪= |𝚽𝚽𝑪𝑪𝑪𝑪|𝟐𝟐。我推导了𝚷𝚷𝑪𝑪𝑪𝑪的解析表达式,它表明非零协方差预测了经典禁区中的不对称动量密度。这为我们提供了一种控制措施,可以使用适当的协方差在经典禁止区域中准备优选的动量分布,但代价是由于不确定性原理(随着粒子的构型空间减小)而产生更大的动量不确定性。另外,由于动量密度是由动量测量统计实验得到的,所以我们可以通过与预测动量分布的比较来测量协方差,也可以间接检验等效原理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Momentum Distribution in the Classically Forbidden Region of a Ballistic Particle at the Turning Point
A wave packet 𝚿𝚿(𝒙𝒙, 𝒕𝒕) of a single particle has a statistical correlation between its position x and momentum p, quantified as the position-momentum covariance. The covariance influences wave packet spreading and the probability current through a given point. This paper shows another effect of the covariance: non-zero covariance can manifest as an asymmetry of the regional momentum density. Consider a selective measurement |𝚿𝚿⟩ → |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ where the initial state |𝚿𝚿⟩ is projected into the state |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ within a smaller region 𝒙𝒙𝟏𝟏 < 𝒙𝒙 < 𝒙𝒙𝟐𝟐. The momentum representation of the projected state is 𝚽𝚽[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] = ⟨𝒑𝒑|𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ and the corresponding momentum density |𝚽𝚽 [𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] | 𝟐𝟐 is the regional momentum density. This paper examines a molecule-sized particle under uniform gravity (represented by a Gaussian wave packet 𝚼𝚼𝟎𝟎 ) at the classical turning point. I consider the effect of the covariance of 𝚼𝚼𝟎𝟎 on the regional momentum density in the classically forbidden region. The initial state 𝚼𝚼𝟎𝟎 with a given covariance is projected into the classically forbidden region, producing the state 𝚼𝚼𝑪𝑪𝑪𝑪. If the corresponding momentum wave function is 𝚽𝚽𝑪𝑪𝑪𝑪, the regional momentum density is 𝚷𝚷 𝑪𝑪𝑪𝑪 = |𝚽𝚽𝑪𝑪𝑪𝑪 |𝟐𝟐. I derive an analytic expression for 𝚷𝚷 𝑪𝑪𝑪𝑪 that shows that a non-zero covariance predicts an asymmetric momentum density in the classically forbidden region. This gives us a measure of control in preparing a preferred momentum distribution in the classically forbidden region using the appropriate covariance, at the price of a larger momentum uncertainty due to the uncertainty principle (as the configuration space of the particle is decreased). Also, since the momentum density is obtained experimentally from the statistics of momentum measurements, we can measure the covariance through comparison with the predicted momentum distribution, as well as indirectly test the equivalence principle.
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来源期刊
Philippine Journal of Science
Philippine Journal of Science Multidisciplinary-Multidisciplinary
CiteScore
1.20
自引率
0.00%
发文量
55
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