Classical statistical pressure is calculated by averaging 2pv where 2p is the momentum change related to an impulse hit Integral F dt (i.e. the particle bounces back) and v is related to flux which in turn is linked to time or the number of particles with momentum p in a certain volume which strike a surface in a given time interval. The time units in the flux term v cancel dt time units in the impulse. We have argued in previous notes that within a quantum wavelength hbar/p (from exp(ipx)) there is no internal time (although there is an external one linked with x=p/m t.). The two dimensions of exp(ipx) account for the direction of motion and so time is not needed internally. [Given that exp(ipx) contains p which in turn is linked to v, there is an external time in the picture, but not an internal one, we argue.] The quantum bound state problem is statistical, but there is only one particle and no time. Thus instead of considering an average of pv, one may consider an average of the impulse i.e. average momentum given by -2idW/dx / W where W is the wavefunction.(A bound state wavefunction is real so the i factor indicates that the momentum is not associated with the entire system translating i.e. it is a bound state.) Impulse, however, is Integral F dt in classical physics, so an average 2p treated as an average impulse contains some kind of average time term. We argue that this seems to be hbar/E in a quantum system. In order for the quantum system to mimic a Maxwell-Boltzmann gas, we argue that the force -dV/dx at x should match hbar/E * average impulse at x. There is no spatial density in W(x) (W*(x)W(x) is spatial density) so one does not need to consider a little box, as in the classical statistical mechanical case. Interestingly, there is a quantum case in which this situation arises, namely the oscillator ground state.
Classical statistical pressure is calculated by averaging 2pv where 2p is the momentum change related to an impulse hit Integral F dt (i.e. the particle bounces back) and v is related to flux which in turn is linked to time or the number of particles with momentum p in a certain volume which strike a surface in a given time interval. The time units in the flux term v cancel dt time units in the impulse. We have argued in previous notes that within a quantum wavelength hbar/p (from exp(ipx)) there is no internal time (although there is an external one linked with x=p/m t.). The two dimensions of exp(ipx) account for the direction of motion and so time is not needed internally. [Given that exp(ipx) contains p which in turn is linked to v, there is an external time in the picture, but not an internal one, we argue.] The quantum bound state problem is statistical, but there is only one particle and no time. Thus instead of considering an average of pv, one may consider an average of the impulse i.e. average momentum given by -2idW/dx / W where W is the wavefunction.(A bound state wavefunction is real so the i factor indicates that the momentum is not associated with the entire system translating i.e. it is a bound state.) Impulse, however, is Integral F dt in classical physics, so an average 2p treated as an average impulse contains some kind of average time term. We argue that this seems to be hbar/E in a quantum system. In order for the quantum system to mimic a Maxwell-Boltzmann gas, we argue that the force -dV/dx at x should match hbar/E * average impulse at x. There is no spatial density in W(x) (W*(x)W(x) is spatial density) so one does not need to consider a little box, as in the classical statistical mechanical case. Interestingly, there is a quantum case in which this situation arises, namely the oscillator ground state.
In Part I, we considered the ideal gas law P= density(x) RT of a Maxwell-Botlzmann gas, which follows from the notion of impulse 2p multiplied by v (flux) and then averaged,and examined how it balanced (on two sides of a box) a force -dV(x)/dx multiplied by density. We argued that in a quantum bound state there is no notion of internal time and hence no flux and so considered an average impulse 2p multiplied by an overall internal time in the system, namely hbar/E. We found that this scenario balanced with -dVd/x matches the quantum oscillator ground state. In this note we ask: What about other potential V(x) situations? Quantum mechanics seems to be a mixture of statistical mechanics and Newtonian dynamics i.e. the time-independent Schrodinger equation is, on average, an energy conservation equation for all energy levels i.e. KE(x)+V(x) = En where KE = {Sum over p a(p)pp/2m exp(kpx)}/{Sum over p a(p)exp(ipx)}. In (1) we took d/dx to bring out a force term -dV/dx and showed if is linked to <ppp/2m> and <pp/2m> , but analyzed these in terms of fluxes. The notion of flux makes sense using external time, but internally there is no time and so here we try to examine the situation in terms of various impulses.
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas.
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
A classical free particle has a specific momentum p and kinetic energy pp/2m. In an ideal gas with no potential P(p)=P(-p) so p average=0 at each x. A quantum free particle has a wavefunction exp(ipx), a specific momentum of p and a kinetic energy of pp/2m. A special feature seems to be that exp(ipx) is associated with a length proportional to 1/p which repeats itself. We argue that this length is associated with the motion p in one dimension together with the two dimensionality. In other words, motion is not a simple translation. If a particle is placed in an infinite well of 1 dimension and length L, then low bound states have a special length (wavelength) of the order of the box. In other words one does not know at which point the particle is located or whether it has already reflected. Thus we argue that from a probabilistic point of view one must include p and -p together. These have different probabilities exp(ipx) and exp(-ipx) because this is a dynamic probability. The interference of the two removes one dimension exposing the periodic cos(px) like nature. If one calculates pp/2m average using -1/2m d/dx d/dx W / W (where W is the wavefunction) one obtains pp/2m (the classical result) because both p and -p have the same magnitude. The dynamic probability, however, does not have the same value for each. Thus p average = -i dW/dx / W is not zero and this result still holds if one multiplies by W*W. In other words there at each x there seems to be more weight to p or to -p. This suggests that an average acceleration may appear in fact: d/dx { -dW/dx / W ) = d/dx d/dx W / W - (dW/dx / W)(dW/dx / W). The quantum bound scenario is a statistical one with an inherent physical (we argue) kind of motion linked to a wavelength. If a particle is knocked out of bound state it has a wavefunction exp(ipx) and a momentum p. It is no longer in a hybrid inteferring statistical state. The modulus of exp(ipx) is 1 at each x as well. Furthermore P(p)=P(-p) = a*(p)a(p)=a*(-p)a(-p) which is not the same result as exp(ipx) versus exp(-ipx) for the dynamical probability. Given P(p)=P(-p) and no knowledge of x one might guess that one has a constant spatial density classical scenario, but this is not the case. As a result a special type of measurement is needed to discern W*(x)W(x) it seems, but this picture of humps and low probability regions is also in keeping with an average acceleration we argue.