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.