Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e. related to {Integral dp p delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.
Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e. related to {Integral dp p delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.
In classical physics, a potential V(x) acts on a particle at a location x regardless of its velocity which is taken to be constant within dx (even though the particle accelerates). Thus writing potential energy V(x) at a given x is like taking a snapshot of the particle when it is at x. The snapshot is static and does not indicate the dynamics of the problem as the particle may be moving either to the right or left, which is irrelevant in terms of the exact potential energy and even kinetic energy at point x. A quantum particle moving with constant velocity is described by a two-dimensional dynamic probability exp(ipx), with cos(px) representing a static density and sin(px) a time-shifted density to capture one directional motion such that exp(ipx)exp(-ipx)=1, we argue. V(x), however, only acts on the static density (linked to cos(px) and not sin(px)). Thus it seems that bound state quantum mechanics needs to remove the dynamic pieces of free particle wavefunctions exp(ipx) through interference. One might argue that for wavelengths of the order of the system length one does not know if one has exp(ipx) or exp(-ipx) at an x point, but for high energy levels, wavelengths become small with respect to the system length and yet interference is still present. Thus it seems that for the calculation of energy one requires a static average snapshot of the bound particle. Physically, we argue that the dynamic piece is still present i.e.. if one knocks out a bound particle and it is found in the p momentum state, it is represented by exp(ipx). Thus the purpose of the bound state wavefunction is to facilitate the calculation of average energy which is associated with a static snapshot type of density and not necessarily represent all of the dynamical information in the system i.e. the two dimensionality of each exp(ipx). As a result, the bound wavefunction does not show motion to the right and then to the left in the high energy limit.
In (1), we argued that classically there are two pictures. The impulse picture delta p = Integral dt F does not distinguish between different velocities if p is the same. In other words, p=m1v1=m2v2 (so time is removed) and v1 and v2 create the same impulse (and quantum mechanically the same interference/diffraction pattern) even though v1 and v2 are physical and different. The second classical picture is the kinetic energy-work one which depends explicitly on time i.e. v dp = Fdx. The two pictures are consistent, but for the one dimensional reflection-refraction problem, a photon count and photon energy balance equation based on time are not independent. Thus one must go to the time-removed impulse picture in order to have two equations, based on the probability exp(ipx), in order to solve the problem. We concluded in (1) that an impulse picture with time removed is necessary to describe interactions in a quantum system in a steady state scenario. Thus this picture should be applied to a bound state. A quantum bound state, however, makes use of a potential V(x) which is associated classically with a kinetic energy at x. Kinetic energy depends on time so how is this described in the impulse picture? We argue that one creates an average of pp based on the impulse probability exp(ipx) a(p) (a(p) is a weight for p) which has time removed. It is the impulse probability exp(ipx) which allows for interference to occur. As a result, <pp> does not have time present and may include p and -p values which classically would occur during different time intervals. Once <pp> is computed in the time-removed picture, however, one divides by 1/2m (in the nonrelativistic case) to create kinetic energy. Thus prms(x), a mathematical average, may be associated with the idea of motion in space and time. pmrs(x), however, may not be considered physical as it is the impulses and exp(ipx) which are physical. As argued in (1), however, classically the two pictures should be consistent. As a result, for high energy levels, the description of <pp> in terms of an impulse picture which does not include time, but then adds 1/2m at the end should be equivalent to a picture in which a particle moves either to the left or right. We suggest that at high energy levels, one may actually assign prms(x) as a physical quantity within a tiny dx and follow the particle in time. Even in the time-removed impulse picture, there is velocity i.e. motion in time. It is just not used to calculate interaction averages in a steady state scenario. We suggest that to move from the quantum picture to the classical, one switches formalisms i.e drops the impulse- time removed picture and uses the time dependent motion in one direction. These two pictures are consistent at the classical level i.e. the classical time picture is a good approximation to the detailed impulse time removed picture which is now occurring in tiny dx regions.
In (1) the Wigner transform of an oscillator is calculated and the high n energy level limit taken using the WKB approximation yielding a result proportional to delta(p - p(x)) + delta(p+p(x)) where p(x) = sqrt(2m(E-.5kxx)). The Wigner transform involves quantum wavefunctions which for bound systems include +/- p components which interfere i.e. a(p)(exp(ipx) + exp(-ipx) for a(p)=a(-p). Classically, a bound particle moves to the right during one time interval and to the left during the other so there is no interference of +/-p. The quantum wavefunction retains +/-p interference for all energy levels and so does not portray the full physical information of the bound system i.e. leads to the two delta functions. For high n (energy level), crests and troughs are very narrow compared to the length of the system, approximating dx units of length. Thus, uncertainty in time is only within these tiny crests/troughs, otherwise there is a distinct time position relationship as in classical physics, but the wavefunction W(x) does not show it. How then may the wavefunction be interpreted? We argue the time independent wavefunction, say for a bound state, represents an averaged steady stream picture which is often not the physical picture. For example, a bound particle is not a steady stream average nor is a finite number of photons hitting an interface of media with different indices of refraction/reflection. Nevertheless, the steady stream approach with time removed may be used to solve the problem i.e. ultimately find a one dimensional spatial density (which averages out the original two dimensional probability of exp(ipx), energy levels etc. For instance, the energy levels of a bound system do not depend on following the particle in time. We raise this issue because we think it may have implications on the calculation of entropy in bound single particle quantum systems. In general, a spatial Shannon’s entropy is defined as: - W(x)W(x) ln[W(x)W(x)] (one dimensional problem). W(x) (wavefunction) is real because p/-p pieces interfere in a steady stream picture. A single bound particle, however, is not a steady stream picture. exp(ipx) is a dynamic probability equal to cos(px)+i sin(px) to indicate the direction of motion. Simply because exp(ipx)+exp(-px) removes the complex portion (i.e. for a steady state picture with a(p)=a(-p)) does not mean that this represents the full physical picture especially as energy levels become high and crests/troughs become narrower with respect to the overall system length it seems there is no justification for the steady stream interference picture.. In other words, using a real W(x) in a Shannon’s entropy expression treats one dimensional probability W(x) = bound wavefunction, but this is not the full probability in the picture, we argue. Full probability is two dimensional and demonstrates direction for many energy levels (especially high ones). For a free particle exp(ipx), cos(px) and sin(px) are essentially the same distribution. Thus cos(px)cos(px) describes the probability distribution in x. For W1(x)=Sum p>0 a(p)exp(ipx), Real(W1(x)) is not equal to Imaginary(W1(x)) because one does not have a simple shift of the real distribution to obtain the imaginary one. The real distribution is chosen to represent probability because the interference of exp(ipx), exp(-ipx) terms removes the imaginary piece, but it is removed only on average. If physically exp(ipx) and exp(-ipx) should not interfere, then one may need to take some kind of average of Real(W1(x))Real(W1(x)) and Imaginary(W1(x))Imaginary(W1(x)) and use this in Shannon’s entropy.
A classical bound state does indicate the direction of motion if one only considers kinetic energy pp/2m. It is the vector quantities, p and x, which indicate direction. In quantum mechanics, a free particle interaction probability (for impulse hits) is directional, i.e. exp(ipx) not= exp(-ipx). Unless one has a single value of pp/2m, KE+= sum over p>0 a(p)exp(ipx) pp/2m / Sum over p>0 a(p)exp(ipx) is not equal to KE- even if a(p)=a(-p). In order to use conservation of energy in quantum mechanics, one is forced to consider both a(p)exp(ipx) +a(p)exp(-ipx) with a(p)=a(-p) which is the correct statistical weighting for a bound state. One may note that exp(ipx) is a probability for interaction and is separate from x=p/m t even though the notion of this velocity is tied to the fact that nonrelativistic kinetic energy is pp/2m. As a result, one must use an average scenario for calculating kinetic energy which seems to be tied to an ensemble average. Such a notion should hold for all energy levels. If one considers the classical limit, then one has p(x)p(x)/2m in a tiny dx. If one were to simply use 1/sqrt(p)exp(ip(x) x)/sqrt(dx) in dx, then Integral W*W dx in dx = C/p. The relative weight of two adjacent dx regions is: p1/ p2 = p1 / (p1+dp). This is equivalent to probabilities proportional to dt= dx/v, where p=mv. Thus, the area represented by W*W captures the number of possible interactions in dx if attempted hits have a fixed period. We argue, however, that given the bound state conservation recipe, one must use a(p) (exp(ipx)+exp(-ipx)) to describe a dx (with p=p(x)), even in the high energy limit. exp(ipx) is simply an interaction pattern (and is a probability) and so if the particle is bound, the correct statistical thing to do for an ensemble seems to be to combine exp(ipx) with exp(-ipx). We then show that quantum mechanics lowers the number of interactions in an area with increased speed more than the classical case. In particular, we find instead of p/(p+dp), 1- dp/p - 2dp/(pp).
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
Steady state scenarios exist in classical physics and even though there is motion (flows) (i.e. changes in time) overall the system does not change in time. Such systems are usually not described in terms of probability. Furthermore, a single particle moving in one direction is described by x(t), i.e. x=vt, so there is no probability and no steady stream scenario.Such a particle, however, may give rise to a steady stream scenario if it encounters a V1-V2 discontinuity (V1, V2 are two constant potentials) and E (energy) remains constant. In such a case, there is a probability to reflect or to pass into the V2 region with a different p, but the same E. This situation automatically introduces probability and may be associated with a steady stream picture, even in classical physics. As a result, classical physics is not as deterministic as it may seem. If probabilities arise at the discontinuity, one may ask: Is there a steady-stream (flux) type of probability associated with a moving particle? In a steady stream situation there is no change in time, but a moving particle clearly changes in time x(t). How does one associate probabilities which do not contain time to flow problems which clearly contain motion? We investigate these questions and try to argue that the quantum exp(ipx) form and the Born W*(x)W(x) wavefunction picture are associated with these questions.