Philippine Journal of Science
152 (3): 931-943, June 2023
ISSN 0031 – 7683
Date Received: 30 Nov 2022

Momentum Distribution in the Classically Forbidden Region
of a Ballistic Particle at the Turning Point

Anthony Allan D. Villanueva*

Institute of Mathematical Sciences and Physics, University of the Philippines Los Baňos
College, Los Baňos, Laguna 4031 Philippines

*Corresponding author: advillanueva1@up.edu.ph

[Download]
Villanueva A. 2023. Momentum Distribution in the Classically Forbidden Region
of a Ballistic Particle at the Turning Point. Philipp J Sci 152(3): 931–943.
https://doi.org/10.56899/152.03.14

 

ABSTRACT

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[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩ where the initial state |𝚿𝚿⟩ is projected into the state |𝚿𝚿[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩ within a smaller region 𝒙𝒙𝟏𝟏<𝒙𝒙<𝒙𝒙𝟐𝟐. The momentum representation of the projected state is 𝚽𝚽[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]= ⟨𝒑𝒑|𝚿𝚿[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]⟩ and the corresponding momentum density |𝚽𝚽[𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐]| 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 𝚷𝚷𝑪𝑪𝑪𝑪=|𝚽𝚽𝑪𝑪𝑪𝑪|𝟐𝟐. I derive an analytic expression for 𝚷𝚷𝑪𝑪𝑪𝑪 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.