In this note, we argue that the Schrodinger equation for one particle represents ensemble averages taken from stochastic momentum values of a single particle at different times, with time being removed. A new time, representing the evolution of the ensemble is then introduced especially for the case in which energy is being added or removed from a system (e.g. V(x,t)). The momentum stochastic features arise from the potential (or force) acting in a stochastic manner leading to a distribution of momentum values at x (at different times). If V(x,t), we argue the system adds or removes some average energy at x at t. We suggest that in reality, stochastic amounts of energy are added or removed (as long as the average is obtained). This leads to a second distribution, namely one in energy being established.. Thus, W(x,t) may be expanded in terms of exp(ipx) for a fixed t or in exp(i Ei t) for a fixed x. As a result, one may calculate average energy using either Average (p*p/2m) + V(x,t) or [Sum over Ei exp(i Ei t) Ei ] / W(x,t). Equating the two stochastic approaches yields the Schrodinger equation.
In this note, we argue that the Schrodinger equation for one particle represents ensemble averages taken from stochastic momentum values of a single particle at different times, with time being removed. A new time, representing the evolution of the ensemble is then introduced especially for the case in which energy is being added or removed from a system (e.g. V(x,t)). The momentum stochastic features arise from the potential (or force) acting in a stochastic manner leading to a distribution of momentum values at x (at different times). If V(x,t), we argue the system adds or removes some average energy at x at t. We suggest that in reality, stochastic amounts of energy are added or removed (as long as the average is obtained). This leads to a second distribution, namely one in energy being established.. Thus, W(x,t) may be expanded in terms of exp(ipx) for a fixed t or in exp(i Ei t) for a fixed x. As a result, one may calculate average energy using either Average (p*p/2m) + V(x,t) or [Sum over Ei exp(i Ei t) Ei ] / W(x,t). Equating the two stochastic approaches yields the Schrodinger equation.
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).
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
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).
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)
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.
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.