If Δxt is the standard deviation of the position probability distribution at time t of a particle moving freely in one dimension, and Δv is the standard deviation of its velocity distribution, it is shown for a suitable choice of time origin that (Δxt)2 = (Δx0)2 + (Δv)2t2 whether the particle is moving classically or quantum mechanically. The quantum-mechanical proof is shown to exactly parallel that in the classical case. In the classical case, the correlation coefficient between the velocity and position distributions is shown to tend to −1 as t → −∞, to increase steadily with time, and to become +1 as t → +∞. A quantum-mechanical analog of this result is also proved.
If Δxt is the standard deviation of the position probability distribution at time t of a particle moving freely in one dimension, and Δv is the standard deviation of its velocity distribution, it is shown for a suitable choice of time origin that (Δxt)2 = (Δx0)2 + (Δv)2t2 whether the particle is moving classically or quantum mechanically. The quantum-mechanical proof is shown to exactly parallel that in the classical case. In the classical case, the correlation coefficient between the velocity and position distributions is shown to tend to −1 as t → −∞, to increase steadily with time, and to become +1 as t → +∞. A quantum-mechanical analog of this result is also proved.
For a wavepacket representing a particle subject to a conservative force consideration is given to the evolution in time of the mean position and momentum and to the evolution of the spread of these quantities as measured by the mean square deviation from the mean. A closed system of equations involving these spreads is obtained by expanding the potential in powers about the mean position and by neglecting terms of third order or higher in the deviations from the mean, an approximation appropriate when the force does not vary too much over the width of the packet. These equations are solved in terms of the trajectories of a classical time-dependent oscillator. These trajectories can be found by differentiation of the trajectories for the force under consideration. In more than one dimension, or for more than one particle, the appropriate generalisation of the spread is the set of second-order correlations.
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
We investigate the transition from quantum to classical mechanics using a one-dimensional free particle model. In the classical analysis, we consider the initial positions and velocities of the particle drawn from Gaussian distributions. Since the final position of the particle depends on these initial conditions, convolving the Gaussian distributions associated with these initial conditions gives us the distribution of the final positions. In the quantum scenario, using an initial Gaussian wave packet, the temporal evolution provides the final wave function, and from it, the quantum probability density. We find that the quantum probability density coincides with the classical normal distribution of the particle's final position obtained from the convolution theorem. However, for superpositions of Gaussian distributions, the classical and quantum results deviate due to quantum interference. To address this issue, we propose a novel approach to recover the classical distribution from the quantum one. This approach involves removing the quantum interference effects through truncated Fourier analysis. These results are consistent with modern quantum decoherence theory. This comprehensive analysis enhances our understanding of the classical-quantum correspondence and the mechanisms underlying the emergence of classicality from quantum systems.
In classical mechanics a constant velocity may be defined by dx/dt where x is in the direction of motion. Thus two dimensions are required and the concept of positive/negative x. Quantum mechanics of a free particle involves a v=x/t, but also independent periodic fluctuations of x and t i.e. exp(-iEt) and exp(ipx) may be considered separately. One, however, cannot lose information we argue so two dimensions are still required to describe motion only using x as well as a concept of negative and positive. (The notion of “only using x” applies to a bound problem.) If the two dimensions only depend on x, then there is no a priori dimension which may be strictly positive such as t. Both dimensions should allow for positive and negative values so that somehow overall the direction of velocity may be defined. Quantum mechanics is concerned with probabilities, but classical probabilities do not introduce negative values. As argued in previous notes, it is possible to have two layers of probabilities P(x) (classical) and P(p/x) linked to exp(ipx) (quantum). The second result contains the two required dimensions and the negative/positive quality. As argued in (1) a quantum free particle has a velocity v=x/t, but the classical action = -Et+px is unchanged if t=1/E and x=1/p. Thus we argue that there may be periodic fluctuations about the classical motion v=x/t. In other words there are uncoupled t and x fluctuations which are linked to p (i.e. v) and are systematic and periodic. These define spatial density. v=x/t defines velocity and has two dimensions and the notion of positive and negative x. We argue that the fluctuation scheme must also define velocity and so requires two dimensions and a notion of positive/negative. As a result the mathematical result of a wavefunction component cos(px) or sin(px) taking on a negative value accounts for information which cannot be lost. For a particle with a single p in a box with infinite walls, a positive or negative hump represents the same probability (sin(px)sin(px) C), but one does not account for the direction of motion. Even using one dimension cos(px) as in a bound state problem, the information of negative and positive values must be retained (in cos(px) over x) because the negative/positive property is used to define overall velocity (even if it is 0 on average for a particle in a box). Thus the concept of interference arises because one cannot lose information in the quantum probabilistic scheme. As noted before it seems one defines new “density” and kinetic energy dimension exp(ipx) and pp/2m exp(ipx) etc.
Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.
Abstract The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.