The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.
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 consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
We propose that the Schrodinger equation results from applying the classical wave equation to describe the physical system in which subatomic particles play random motion, thereby leading to quantum mechanics. The physical reality described by the wave function is subatomic particle moving randomly. Therefore, the characteristics of quantum mechanics have a dual nature, one of them is the deterministic nature carried on from classical physics, and the other is the probabilistic nature coined by particle's random motion. Based on this model, almost all of open questions in quantum mechanics can be explained consistently, which include the particle-wave duality, the principle of quantum superposition, the interference pattern of double-slit experiments, and the boundary between the classical world and the quantum world. The current quantum mechanics is a mixture of matrix mechanics and wave mechanics, which are sharply conflicting in principle. Matrix mechanics treats quantum particles as classical particles with fixed relation between the particle's position and its momentum. The matrix mechanics, in fact, belongs to the old quantum theory. Both Born's non-commutative relation and Heisenberg uncertainty relation originate from matrix mechanics. However, in wave mechanics, there is no any fixed relation between the particle's position and its momentum, and the particle's position and its momentum belong to immeasurable physical quantities. Therefore, there is no need for non-commutative relation and uncertainty relation in wave mechanics.
There seems to be much interest in the literature (1) in associating quantum mechanics (in particular the Schrodinger equation) with a classical joint phase space probability F(x,p,t) such as appears in classical Liouville theory etc. Here, we consider that P(x)dx = dx/L for a particle at rest and analyze the extra physics which appears when one considers a single particle bouncing back and forth in a box, which is what a person in a frame moving with -v would see (albeit with a Lorentz contracted L.) We suggest that the constraint L is important as it is linked with physical forces which appear when the particle moves. In other words, a particle at rest is linked with a static probability, while for motion, one must have a probability which is linked to more than x and t, but to force considerations in x and t, a dynamical probability. (Experiment bears this out as we discuss.) In particular, we argue that in the case of a moving particle, two physical concepts related to force appear, namely pressure which is an average over time and impulse hit which is only linked with space. Thus, two different particles may have the same p, but different v and pressure values. We thus suggest a physical grouping of (p,x) and (pressure, t). We then note that one needs to somehow preserve the notion of P(x)=1/L, i.e motion does not introduce a bias to x, nor to time. In a rest frame, there is no motion and so there is neither p nor pressure, but P(x)=1/L pertains. If one tries to define a probability with two groupings, one for pressure, t and the other for p,x, then if one multiples a p1 case by a p2 case, the p,x grouping should yield p1+p2, x. Furthermore, a p,x grouping should imply mixed terms, e.g. px etc, otherwise one does not really have an association between the two variables. This suggests a pf(x) form linked to the p-x grouping. Given the notion of special relativity, it seems that one may replace pressure with the notion of energy and create a Lorentz invariant probability. This would then lead to the free particle quantum wavefunction (probability) exp(-iEt+ipx), but we argue that the roots are in classical physics and a desire to physically describe the physical information present for a particle bouncing back and forth in a box.
In (1) it is shown how the Schrodinger equation can be “forced” in a sense to follow from the Fokker-Planck classical stochastic equation which contains a classical probability density. Other approaches in the literature which are similar use continuity equations and also a classical density. The key to all of these approaches is to assume, in an ad hoc manner it seems, that classical probability is equivalent to W*(x)W(x) where W(x) is a complex function called the wavefunction. In principle, one may write a density as W*(x)W(x). This is mathematically correct even if, for the sake of argument, it is ad hoc. The problem is that for a constant density (i.e. constant momentum and speed), one does not know if W(x)=exp(ipx) or W(x)=exp(-ipx). We consider here the arguments shown in (1). If a free particle has a density(x)=1/L, it may be associated with W(x) = exp(ipx)/sqrt(L). A free particle density, however, is associated with constant p (momentum) and assumes one knows x(t). If one considers the free particle to be trapped in a one-dimensional box with infinite potential walls, it travels to the right with p and then to the left with -p. The density(x) classically is 1/L, where L is the length of the box. The density is constant in time, but the direction of motion is a function of time. Thus, information is lost considering an equation such as the Fokker-Planck equation which uses a density function which does not track the direction of motion in time. In particular, writing density(x) = W*(x)W(x), which is done in (1) and applying this density to a bound particle in a box with infinite walls leads to an equation linear in W(x). Thus exp(ipx) and exp(-ipx) are both possible solutions if V(x)=0. Given that the conservation type equation obtained from the Fokker-Planck (or continuity) equation is linear in W, exp(ipx)+exp(-ipx) is a solution. Thus there is interference which leads to a density which does not match a constant classical density which should apply to a particle moving to the right and then to the left in a bound state because one assumed a classical picture a priori. Simply suggesting that density classical = W*(x)W(x) mathematically does not mean that the a priori classical picture may be dropped. As a result, we argue that there exists a contradiction in approaches which write classical density as W*(x)W(x) and then obtain the Schrodinger equation. As a remedy, we suggest that W(x) does not follow from a theory of classical density=W*(x)W(x), but vice versa. One needs equation which deal with W(x) a priori, and once it is found, one may create a classical density through W*(x)W(x).