Addendum Feb. 4, 2021 In this note, a potential of .5kxx + f(t)x is considered together with a product wavefunction Wa(y,t)exp(i y db/dt) where y=x-b(t). Product wavefunctions when inserted into the Schrodinger equation lead to two equations. The first may be chosen to depend only on one product e.g. Wa. In this case, it is a harmonic oscillator Schrodinger equation with an extra c(t)Wa term which may be handled by a pure time phase i.e. exp(i Integral (0,t) c(t1)dt1)). The second equation incorporates the potentail f(t)x and involves a coupling term -1/m dWa/dy d/dy exp(i y db/dt). This term, however, is exactly canceled by a term from i d/dt (partial) Wa(y-b(t)) exp(i y db/dt) so the coupling is eliminated. Thus, one has a second differential equation in b(t) linked to the potential f(t)x or f(t)y- f(t)b(t) which is completely uncoupled. Thus, there are two decoupled types of motion. Harmonic motion in y=x-b(t) and collective motion related to b(t) linked directly to f(t). Addendum Feb. 3, 2021 The term -1/2m d/dy d/dy W1 in ((6)) is db/dt db/dt (1/2m) and is incorporated in c(t)Wo in equation ((5)). Also i d/dt (partial) {exp(i(x-b/m)db/dt) W(x-b/m)) = -i/m dW/dy db/dt +(1/m)db/dtdb/dt -yd/dt db/dt as seen in the equation directly below ((6)) on the LHS. An extra pure time piece is added i.e. +(1/m)db/dtdb/dt. The “time-independent” Schrodinger equation appears to be directly related to stochastic motion, we argue. If there is no potential, the wavefunction exp(ipx) is statistical, indicating periodic motion with a norm of one for all x. If a time-independent potential is added, it may be decomposed as V(x)=Sum over k Vk exp(ikx). Thus, time is present not just in exp(-iEt), but in the stochastic behaviour of the potential which is only V(x) on average. At each time, one does not know which Vk acts. Thus, the time-independent Schrodinger equation could perhaps be called the purely stochastic Schrodinger equation. It is possible to have a time dependent Schrodinger equation with a time dependent potential. Sometimes the time in the potential is hidden by a variable change as the equation involves partial derivatives. We try to argue that this potential contributes to collective motion. Time is already present in V(x), but in a stochastic manner. Explicit time (or a time related variable change) means a lack of stochasticity as one follows the potential in a predictable manner in time. It is possible that collective motion may be associated with a reference frame. For example, one may remove center-of-mass motion and consider only internal motion i.e. the quantum stochastic internal behaviour. In this note, we briefly examine two examples. The first is a free quantum particle viewed from a constantly accelerating frame as discussed in (1). In such a frame, the free particle appears as a particle in a gravitational potential +mgx1, but x1= x-X(t) so X(t) suggests collective motion. One may argue that x represents stochastic motion, but this is not necessarily the case because one may have collective potential energy linked to collective kinetic energy. In other words, id/dt partial W may yield a term which cancels mgx indicating that it does not act as Sum Vk exp(ikx). The second example involves a quantum oscillator with an extra xf(t) potential. We argue that this leads to collective motion as the problem may be recast in terms of a variable y=x- b(t) and solved as a “time-independent oscillator” in y as discussed in (2). Again, there exists the notion of a kind of collective motion or extra frame which leads to a phase factor exp(iy d/dt partial b(t)) as the stochastic quantum behaviour defines the physical density and the phase term exp(i phase) disappears from the density). In such a case, W(y) the time-independent oscillator solution may be written as: Sum over p a(p) exp(ipy), but y contains t so the statistical nature of exp(ipy) is somewhat altered frome exp(ipx) i.e. there is an extra time related phase exp(i p f(t)) which has the appearance of a kind of collective motion as all p values are in synch with the same f(t) which may be sin(wt) for example.
Addendum Feb. 4, 2021 In this note, a potential of .5kxx + f(t)x is considered together with a product wavefunction Wa(y,t)exp(i y db/dt) where y=x-b(t). Product wavefunctions when inserted into the Schrodinger equation lead to two equations. The first may be chosen to depend only on one product e.g. Wa. In this case, it is a harmonic oscillator Schrodinger equation with an extra c(t)Wa term which may be handled by a pure time phase i.e. exp(i Integral (0,t) c(t1)dt1)). The second equation incorporates the potentail f(t)x and involves a coupling term -1/m dWa/dy d/dy exp(i y db/dt). This term, however, is exactly canceled by a term from i d/dt (partial) Wa(y-b(t)) exp(i y db/dt) so the coupling is eliminated. Thus, one has a second differential equation in b(t) linked to the potential f(t)x or f(t)y- f(t)b(t) which is completely uncoupled. Thus, there are two decoupled types of motion. Harmonic motion in y=x-b(t) and collective motion related to b(t) linked directly to f(t). Addendum Feb. 3, 2021 The term -1/2m d/dy d/dy W1 in ((6)) is db/dt db/dt (1/2m) and is incorporated in c(t)Wo in equation ((5)). Also i d/dt (partial) {exp(i(x-b/m)db/dt) W(x-b/m)) = -i/m dW/dy db/dt +(1/m)db/dtdb/dt -yd/dt db/dt as seen in the equation directly below ((6)) on the LHS. An extra pure time piece is added i.e. +(1/m)db/dtdb/dt. The “time-independent” Schrodinger equation appears to be directly related to stochastic motion, we argue. If there is no potential, the wavefunction exp(ipx) is statistical, indicating periodic motion with a norm of one for all x. If a time-independent potential is added, it may be decomposed as V(x)=Sum over k Vk exp(ikx). Thus, time is present not just in exp(-iEt), but in the stochastic behaviour of the potential which is only V(x) on average. At each time, one does not know which Vk acts. Thus, the time-independent Schrodinger equation could perhaps be called the purely stochastic Schrodinger equation. It is possible to have a time dependent Schrodinger equation with a time dependent potential. Sometimes the time in the potential is hidden by a variable change as the equation involves partial derivatives. We try to argue that this potential contributes to collective motion. Time is already present in V(x), but in a stochastic manner. Explicit time (or a time related variable change) means a lack of stochasticity as one follows the potential in a predictable manner in time. It is possible that collective motion may be associated with a reference frame. For example, one may remove center-of-mass motion and consider only internal motion i.e. the quantum stochastic internal behaviour. In this note, we briefly examine two examples. The first is a free quantum particle viewed from a constantly accelerating frame as discussed in (1). In such a frame, the free particle appears as a particle in a gravitational potential +mgx1, but x1= x-X(t) so X(t) suggests collective motion. One may argue that x represents stochastic motion, but this is not necessarily the case because one may have collective potential energy linked to collective kinetic energy. In other words, id/dt partial W may yield a term which cancels mgx indicating that it does not act as Sum Vk exp(ikx). The second example involves a quantum oscillator with an extra xf(t) potential. We argue that this leads to collective motion as the problem may be recast in terms of a variable y=x- b(t) and solved as a “time-independent oscillator” in y as discussed in (2). Again, there exists the notion of a kind of collective motion or extra frame which leads to a phase factor exp(iy d/dt partial b(t)) as the stochastic quantum behaviour defines the physical density and the phase term exp(i phase) disappears from the density). In such a case, W(y) the time-independent oscillator solution may be written as: Sum over p a(p) exp(ipy), but y contains t so the statistical nature of exp(ipy) is somewhat altered frome exp(ipx) i.e. there is an extra time related phase exp(i p f(t)) which has the appearance of a kind of collective motion as all p values are in synch with the same f(t) which may be sin(wt) for example.
The time-dependent Schrodinger problem of an oscillator .5kx*x together with the additional potential -x f(t), where f(t) is a function of time (often periodic), has already been solved in the literature (1). The solution involves a variable transformation y= x - b(t), where b(t) is a function of time which will turn out to be related to f(t) and W(x,t)= exp(i y d/dt b(t)) Wosc( y) exp(i Eosc t) . This yields an oscillator type time dependent equation with a term q(t) Wosc(y) which can be removed by using Wosc(y) exp(i c(t)). Wosc( y) is the oscillator wavefunction with the space-time variable y and Eosc the usual oscillator energy levels. Thus, by using the variable transformation together with W(x,t)=Wosc(x)exp(iEosc t) g(x,t) (i.e. a product of factors) one can obtain an “time-independent like” Schrodinger equation in y (after performing one more transformation Wosc(y)exp(i c(t)). In a previous note (2), we argued that if one writes a wavefunction as W(x)=Wo(x)q(x), where Wo(x) is the ground state wavefunction, one obtains a time-independent Schrodinger equation for Wo(x), which contains V(x), and a second eigenvalue equation coupling Wo(x) and q(x) with an eigenvalue equal to E-Eo. In particular, it is the time flux [d/dt W(x,t)]/W(x,t) which becomes linked with q(x) and the spatial flux [d/dx W(x)]/W(x). In (3), we argued that bound state quantum mechanics contains two velocities v(x), the classical velocity and u(x)/m = [d/dx W(x)]/ W(x) and that each behave differently. For example, v(x) behaves according to Newton’s law, while u(x)/m does not (except for the oscillator ground state). In this note, we try to apply these two ideas to the case of the oscillator with the time-dependent extra potential x f(t). We see that writing W(x,t) as a product (after the variable change) leads to two sets equation, one being a time-independent like equation for the oscillator and the other linking u(x)/m to a portion of time flux [d/dt W(x,t) ]/ W(x,t) |x constant. In this case, the time flux is directly related to d/dx [W*(x,t)W(x,t)] i.e. the physical density. Thus, a change in physical density in time is related to a change in physical density in space. Thus, two physical processes, it seems, are occurring. One is the W osc (y) resonance (albeit it a strange one because y=x + b(t)) and the second flux coupling with exp(i b(t) y), part of the oscillator potential and part of the additional potential -x f(t). Given that the problem has been mathematically solved already in the literature (1), our objective is to attempt to see if there is physical meaning in terms of ideas we have presented in previous notes, namely flux, the notion of a product wavefunction leading to a separate resonance type of reaction and double resonances.
In a series of notes (1), we argued that the time-independent Schrodinger equation may be considered as a statistical conservation of average energy equation. We further argued that V(x)=Sum over k Vk exp(ikx) delivers stochastic potential hits which create a momentum distribution W(x)= Sum over p a(p) exp(ipx) at each x. The form exp(ipx) describes a free particle and a statistical distribution is made of such free particles. The form exp(ipx) has a modulus of 1 indicating that P(x)=constant. d/dx exp(ipx) = ip suggesting the change in conditional probability with space is related to classical momentum. Given a momentum distribution, each p at x occurs at a different time. Thus, one does not follow a classical particle in time x(t), nevertheless an average energy conservation equation applies. Taking the time-independent Schrodinger equation E W = -1/2m d/dx d/dx W + V(x) W and writing EW = i d/dt exp(-iEt), one obtains a time-dependent equation. We argue that if the time-independent case represents a classical conservation of average energy, the time-dependent equation should be linked to Newton’s equation of motion. This immediately leads to a problem because we argue one does not want x(t) for a statistical system. As a result, x in the time-dependent Schrodinger equation, which represented spatial position before, now represents a spatial variable that is independent of time. We consider the oscillator case and note that an amplitude is such a candidate where x(t)=XA(t). Thus, the time-dependent Schrodinger equation ultimately becomes an equation in A(t) with the amplitude cancelling, at least in the free particle and oscillator case. From such a consideration, one may work “backwards” and find a function for which the time-dependent Schrodinger equation (equating powers of X) becomes Newton’s second law. From this, one may show that exp(i Action) where Action = Integral dt1 (0,t) L where L is the classical Lagrangian is this solution with d/dX applying to X which is an amplitude, not x(t).
If a Lagrangian contains a potential dependent on dx/dt, it is possible this does not contribute to the classical equations of motion. Nevertheless, the fact that the potential is present suggests there is a physical impact. A different Lagrangian yielding the same equations of motion, does not necessarily represent the same physical situation i.e. the equations of motion do not uniquely determine a physical situation. In particular, using the Lagrange formalism yields momentum=p = dL / d (dx/dt) which may contain the usual m dx/dt term plus terms from velocity related potentials. We argued in previous notes, that for quantum mechanics there is a momentum distribution at each x point which yields an average pp/2m at x. Moving from one x to another yields a change in pp indicating an average acceleration. The potential V(x), it was argued, may be written as Sum over k Vk exp(ikx). Thus, it is stochastic, yielding momentum hits of k which only yield a force -dV/dx on average. Given that average acceleration is stochastic, average momentum should also be stochastic. For the case of a potential A(x)dx/dt, the Lagrangian momentum dL/d (dx/dt)= mdx/dt +A(x). dx/dt is an average of stochastic value i.e. Sum over p a(p) exp(ipx), but A(x) is also stochastic i.e. A(x)= Sum over k Ak exp(ikx). One may combine the two terms to create a term of the form bk exp(i j x). We argue that this contains the combined stochasticity of p and A(x). Special considerations show that if W(x)=Wo(x)W1(x), then one may break out W1(x) as a phase related directly to A(x) i.e. W1(x)= exp{ i (Integral dx1 (0,x) A(x1) } which is experimentally observable.
A single particle bound wavefunction may be written as a Fourier series W(x)exp(-iEt) = exp(-iEt) Sum over p a(p) exp(ipx). We argued in previous notes (1) that exp(ipx) is a statistical free particle conditional probability whose modulus of 1 corresponds to P(x)=constant. For the case of a potential V(x), we argue it is stochastic and write it as: Sum over k Vk exp(ikx). The stochastic potential gives rise to a momentum distribution: P(p/x) = a(p) exp(ipx) / W(x) at each x. A resonance, however, seems to occur because writing W(x) and V(x) in the time-independent Schrodinger equation as Fourier series, yields: pp/2m a(p) + Sum over k Vk a(p-k) = E a(p) ((1)) Thus, a(p) is really a combination of other a(p-k) or, in other words, there is a correlation between different free particle momentum states. If such is the case for single particle interactions with a stochastic potential, we argue it is possible a correlation may exist for two particles. In other words, a stochastic potential may not necessarily produce two separate resonances for two quantum particles such as ((1)), but may instead form an overall resonance involving both momenta p1 and p2 “interacting or communicating” with Vk as in the case of entanglement. We attempt to investigate these ideas in this note.
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
The oscillator propagator exp(i Action) = Q(t) exp[i mw/2 { cos(wt)/sin(wt) (XX + YY) - 2XY / sin(wt)} ] = Q(t) exp( i (-m/2 dA/dt / A) (XX +YY) - (mw/2) 2XY/A ) (where XA(t) is a classical solution with X as amplitude) is a solution of the time-dependent Schrodinger equation. Inserting the form with A(t) into the time-dependent Schrodinger equation yields Newton’s second law (with two other terms which cancel) i.e. m d/dt dA/dt = - k A(t). Newton’s second law may be multiplied by dA/dt and integrated, yielding the well-known conservation of energy equation: Energy = XX dA/dt dA/dt (m/2) + (k/2) XX AA. Thus, the two are equivalent as is known from classical mechanics. One may consider solving the time dependent Schrodinger equation in a different manner, namely postulating a separable solution of the form: exp(-i C t) H(x). Substituting into the oscillator Schrodinger equation yields C = -1/2m d/dx d/dx W + .5 k xx W. Mathematically, this equation may be solved, but only a discrete set of C values yield solutions, namely C= .5 w(n+.5) with corresponding Wn(x). We argue that this mathematical equation may be linked to a physical one, namely the conservation of energy equation if C is associated with energy. In such a case, one must associate -1/2m d/dx dW/dx with classical kinetic energy at a x. At this point, we suggest one must establish a physical interpretation for -1/2m d/dx dW/dx if W. We argue that X in exp(i Action) is an amplitude. Allowing all values of X (e.g. -1/2m d/dX d/dX) means allowing all amplitudes for a particular spring. Thus it seems that one is considering a fluctuating amplitude picture in which these fluctuations can yield an average energy associated with a particular En. This average equation follows classical mechanics in terms of average KE(x) + V(x) = En.