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The Philippine journal of science

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

Anthony Allan Villanueva

Published Mar 21, 2023β€’ 0 citations
Publisher

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⟩ β†’ |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ where the initial state |𝚿𝚿⟩ is projected into the state |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ within a smaller region π’™π’™πŸπŸ < 𝒙𝒙 < π’™π’™πŸπŸ. The momentum representation of the projected state is 𝚽𝚽[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] = βŸ¨π’‘π’‘|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 𝚼𝚼π‘ͺπ‘ͺπ‘ͺπ‘ͺ. If the corresponding momentum wave function is 𝚽𝚽π‘ͺπ‘ͺπ‘ͺπ‘ͺ, the regional momentum density is 𝚷𝚷 π‘ͺπ‘ͺπ‘ͺπ‘ͺ = |𝚽𝚽π‘ͺπ‘ͺπ‘ͺπ‘ͺ |𝟐𝟐. I derive an analytic expression for 𝚷𝚷 π‘ͺπ‘ͺπ‘ͺπ‘ͺ 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.

DOI
10.56899/152.03.14

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A quantum particle (represented by a Gaussian wave packet) under uniform gravity, projected upward (the conventional positive direction) along the vertical degree of freedom x, may have a negative probability current on a set of points q. This imposes the condition that the position and momentum observables of the state are negatively correlated. We calculate the effect of the negative correlation on the individual contributions to the probability current by the positive and negative momentum amplitudes of the state using an integral representation of the probability current. Negative position-momentum correlation causes a transient attenuation of all contributions to the probability flow, and wave packet spreading is temporarily mitigated.
The weak equivalence principle of gravity is examined at the quantum level in two ways.First, the position detection probabilities of particles described by a non-Gaussian wave packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated.These probabilities exhibit mass dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle.Second, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent.Such a mass dependence is shown to be enhanced by increasing the non-Gaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet.The mass dependence of both the position detection probabilities and the mean arrival time vanishes in the limit of large mass.Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter.A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
We discuss a class of solutions of the time-dependent SchrΓΆdinger equation such that the position uncertainty temporarily decreases. This self-focusing or contractive behavior is a consequence of the anti-correlation of the position and momentum observables. Since the associated position density satisfies a continuity equation, upon contraction the probability current at a given fixed point may flow in the opposite direction of the group velocity of the wave packet. For definiteness, we consider a free particle incident from the left of the origin, and establish a condition for the initial position-momentum correlation such that a negative probability current at the origin is possible. This implies a decrease in the particle's detection probability in the region x > 0, and we calculate how long this occurs. Analogous results are obtained for a particle subject to a uniform gravitational force if we consider the particle approaching the turning point. We show that position-momentum anti-correlation may cause a negative probability current at the turning point, leading to a temporary decrease in the particle's detection probability in the classically forbidden region.
Abstract Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes <?CDATA ψ(k,t)=ψ(k)exp⁑(βˆ’iΟ‰kt)\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)ψ(k,t)=ψ(k)exp(βˆ’iΟ‰k​t)?> ψ ( k , t ) = ψ ( k ) exp ( βˆ’ i Ο‰ k t ) yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities β„˜ ΞΌ ( x , t) = (ℏ/2i)[ψ +( x , t)βˆ‚ ΞΌ ψ( x , t) βˆ’ βˆ‚ ΞΌ ψ +( x , t) Β· ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields Ο•( x , t), and therefore the canonical local energy-momentum densities <?CDATA H(x,t)=cP0(x,t){ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)H(x,t)=cP0(x,t)?>  ( x , t ) = c  0 ( x , t ) and <?CDATA P(x,t){ \mathcal P }({\boldsymbol{x}},t)P(x,t)?>  ( x , t ) differ from β„˜ ΞΌ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density <?CDATA H(x,t){ \mathcal H }({\boldsymbol{x}},t)H(x,t)?>  ( x , t ) , the canonical charge density Ο±( x , t), the energy pseudo-density <?CDATA H~(x,t)=cβ„˜0(x,t)\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)H~(x,t)=cβ„˜0(x,t)?>  ˜ ( x , t ) = c β„˜ 0 ( x , t ) , and the Born density ∣ψ(
We derive an inequality that the parameters of a 1D free-particle Gaussian wave packet with a positive group velocity, approaching a given region x > q, must satisfy such that a negative probability current J exists on q. Local probability conservation implies the counter-intuitive result that the particle detection probability in the region x > q is actually decreasing. The condition J < 0 requires the negative correlation of the position and momentum observables of the state, but the time scales for the negative current and anti-correlation regimes are not identical. Using a probability current operator, we obtain an integral representation of J in momentum space for any free particle wave packet. We use this integral representation to distinguish the separate contributions to J by the positive and negative momentum components, and we identify a third contribution to J composed of cross-terms of both momenta. For the specific case of a Gaussian wave packet with a negative correlation between its position and momentum, the positive momentum component can contribute a negative value to the probability current.
Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass mmm falling in a uniform gravitational field ggg, we use this approach to obtain an exact explicit expression for the probability density of the time of arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and xxx is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.