In an effort to describe quantum and classical mechanics with the same language, a wave equation for a continuous transition from quantum to classical mechanics has been proposed. Furthermore, the equivalence of this nonlinear equation with a linear one, known as the scaled equation, which is just the Schrodinger equation with the scaled Planck constant instead of the usual one, has been proved. Using this equation, we'll study three interesting phenomena; these include tunneling through a rectangular barrier, total reflection from a hard wall, and the gravitational weak equivalence principle in quantum, transition and classical regions. Time-independent scaled equation for the stationary states is derived and solved for a flux of particles incident on the barrier. The relations show that tunneling probability is exactly zero in the classical regime. For the other problems, we use a Gaussian wavepacket to calculate the expectation value of the position operator in reflection from the hard wall and to estimate the detection probability and arrival time in the problem of the gravitational weak equivalence.
In an effort to describe quantum and classical mechanics with the same language, a wave equation for a continuous transition from quantum to classical mechanics has been proposed. Furthermore, the equivalence of this nonlinear equation with a linear one, known as the scaled equation, which is just the Schrodinger equation with the scaled Planck constant instead of the usual one, has been proved. Using this equation, we'll study three interesting phenomena; these include tunneling through a rectangular barrier, total reflection from a hard wall, and the gravitational weak equivalence principle in quantum, transition and classical regions. Time-independent scaled equation for the stationary states is derived and solved for a flux of particles incident on the barrier. The relations show that tunneling probability is exactly zero in the classical regime. For the other problems, we use a Gaussian wavepacket to calculate the expectation value of the position operator in reflection from the hard wall and to estimate the detection probability and arrival time in the problem of the gravitational weak equivalence.
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.
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notion of time, where space and time have the same hierarchy, essential in describing events in spacetime. On the other hand, in quantum mechanics, time appears as a classical parameter, meaning that it does not have an uncertain relation with its canonical conjugate. In this work we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space: the time parameter t and the position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ in one subspace, and the position parameter x and time operator T in the other subspace. Time as an operator is better suited for describing tunneling processes. We then solve the 1/2-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that for wave packets well resolved in time, the expectation value of the operator T is the energy average of the classical time ${T}_{\text{class}}=\ensuremath{\partial}S/\ensuremath{\partial}E$, where S is the classical action, which can be real or imaginary. The imaginary classical time does not contribute to the traveling time. Furthermore, we apply our results to a Gaussian energy distribution and compare them to previous works. This work is a correction of a previous paper [Phys. Rev. A 107, 052220 (2023)].
We use the Feynman path integral approach to nonrelativistic quantum mechanics twofold. First, we derive the lagrangian for a spinless particle moving in a uniformly but not necessarily constantly accelerated reference frame; then, applying the strong equivalence principle (SEP) we obtain the Schroedinger equation for a particle in an inertial frame and in the presence of a uniform and constant gravity field. Second, using the associated Feynman propagator, we propagate an initial gaussian wave packet, with the final wave function and probability density depending on the ratio m/hbar, where m is the inertial mass of the particle, thus exhibiting the fact that the weak equivalence principle (WEP) is violated by quantum mechanics. Although due to rapid oscillations the wave function does not exist in the classical limit, the probability density is well defined and mass independent when hbar goes to 0, showing the recovery of the WEP. Finally, at the quantum level, a heavier particle does not necessarily falls faster than a lighter one; this depends on the relations between the initial and final common positions and times of the particles.
The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approaches to the problem. Here, we tackle the problem anew through the time of arrival (TOA) operator approach. This is done by constructing the TOA operator for a nonrelativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA operator is constructed under the constraint that the inertial and gravitational masses are equivalent and that Galilean invariance is preserved. These constraints are implemented by Weyl quantization of the corresponding classical TOA function for the projectile. The expectation value of the TOA operator is explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the WEP with quantum mechanics. The full extent of the violation of the WEP is shown through the mass dependence of TOA distribution for the projectile.
We have argued in previous notes that quantum formalism emerges from the one dimensional photon reflection-refraction scenario which is readily observable in the classical world. What about the situation for a particle with rest mass? In previous notes, we have considered the case of a constant E,p and V1 which changes at xo to E,p2,V2 (all again constants with E remaining unchanged). This problem is then associated with an incident and reflected particle for x<xo and a particle with p2 for x>-xo. In a classical world, however, how does one obtain reflection and refraction together? If there is a strong wall, a particle may reflect with -p and the same kinetic energyE, but it does not pass through the wall. If there is a ramp at xo leading to a horizontal floor a little higher than that for x<xo, then one has a conserved total E, but a new p2 and V2. The ramp, however, is not a sudden change at xo, but occurs over a range of x. As a result, it seems the quantum scenario, rather than leading to the classical case in a high energy limit (as in bound states), is associated with fluctuations and non-sharp boundaries. It is fine to have a sharp wall at x=xo, but in quantum mechanics, one has atoms with electrons moving in space according to probability distributions. Furthermore, there is movement due to temperature. One does not have a sharp edge or ramp scenario alone, but fluctuations which could lead to both. This suggests stochasticity at a point of discontinuity, but also stochasticity in the regions V1 and V2. In other words, V1, V2 and the sharp discontinuity points are all approximations (averages). If one accepts the fluctuations at xo (ramp and sharp wall), then classically one might impose a pressure balance equation at an idealized xo, i.e.: AAp -BBp=p2CC where AA represents a positive number etc. AA represents the incident particle, BB the reflected and CC the particle which moves up the ramp. Nonrelativistically and in the photon case, the pressure balance equation is also equivalent to a conservation of particle number or energy equation. Thus one has two unknowns (if A is known), i.e. B and C, and one equation. Here, however, we wish to stress a different, but related issue. The pressure balance equation allows for multiple solutions because it only exists at xo. For example, BB=0 is a solution as is BB=.5, CC=.5. If p2 is almost equivalent to p, one might expect that BB=0. In other words, there is a specific solution which goes beyond pressure balance, i.e. here seems to be a p (momentum) biased solution. This forces one to go beyond classical mechanical ideas, even if one introduces a fluctuating wall-ramp scenario at xo. The fluctuating scenario suggests fluctuations in V1 and V2 as well, suggesting one may have some kind of dynamical equilibrium in these regions. Would this be enough to create a momentum bias, linked say to a continuity of a function of p,x and its first derivative at xo? This extra momentum bias seems to be the extra feature which emerges in a quantum treatment of the problem, which only occurs because one introduces fluctuations at the xo point. We suggest that a dynamical equilibrium approach solves the problem if one maximizes Shannon’s entropy subject to a the constraint of a constant average classical action written as: -P(t) iEt + ipx P(x) with a Lagrange multiplier of 1. “i” is introduced to a have a periodic/dynamic probability, i.e. one that does not grow. Thus we argue that quantum mechanics for a particle with rest mass emerges because a classical pressure balance equation for a fluctuating wall-ramp picture at a V1-V2 junction allows for multiple solutions. Ifr V1 is almost equal to V2, one has a biased or selective solution which should force BB to be 0. This may be done through continuity, but there must be a physical basis for the function used and we suggest that this is maximization of Shannon’s entropy subject to an average classical action introduced to make the probability periodic due to the dynamical nature of the problem. Thus there is not just stochasticity at the discontinuity point xo, but throughout x.
This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.