In a previous note (1), the Schrodinger time-independent equation is divided by the wavefunction and then a spatial derivative taken. This leads to an equation with force=-d/dx V(x) present. Then, d/dx ( d/dx d/dx W / W) is written microscopically as (Sum over p p3 f(p) exp(ipx)) / W – (d/dx d/dx W)/W (d/dx W)/W. Here W(x) is the wavefunction. Thus, force is related to a microscopic average kinetic energy flux minus an average kinetic energy times an average flux. In this note, we try to examine the same situation in a rotating frame and examine force in this case and also try to consider microscopic features of V(r).
In a previous note (1), the Schrodinger time-independent equation is divided by the wavefunction and then a spatial derivative taken. This leads to an equation with force=-d/dx V(x) present. Then, d/dx ( d/dx d/dx W / W) is written microscopically as (Sum over p p3 f(p) exp(ipx)) / W – (d/dx d/dx W)/W (d/dx W)/W. Here W(x) is the wavefunction. Thus, force is related to a microscopic average kinetic energy flux minus an average kinetic energy times an average flux. In this note, we try to examine the same situation in a rotating frame and examine force in this case and also try to consider microscopic features of V(r).
In a previous note (1), it was shown that by taking the derivative of the time-independent Schrodinger equation in the form: -1/2m (d/dx d/dx W /W) + V(r)=E yields an equation linking various fluxes to–d/dx V i.e. force. In other notes, it was shown the time-independent Schrodinger equation may be written microscopically for each p momentum plane wave as: p2/2m f(p) + Sum j+k=p Vj f(k) = E f(p) ((1)) Here, the same E (energy) applies to all p momentum waves. Thus, it is as if there is a dynamical resonance for each p wave, with various k momentum waves combining with the potential (Vj= the jth Fourier transform of V(r)) to form a p momentum wave. In this note, we wish to examine force microscopically in terms of ((1)) and also to consider the effects of noninertial frames or constrained motion on microscopic force.
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.
The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.
The Schrodinger equation for a bound state includes the classical potential V(r) for which –d/dxj V represents Fj or force in the j direction. It has been argued in a previous note (1), that one may attribute physical meaning to the Fourier series of V(r). In such a case, V(r) and W(r) (the wavefunction) are quantities averaged over momentum (or vectors) as well as being averages over time. One may write the product of V(r)W(r) as one of two Fourier series and then reorganize into a series with basis exp(ikr). In such a case, one examines an interaction with V at each momentum wave level. Then the average is only over time as different k momentum states of in W combine with different p momentum states of the Fourier series of V(r). For a p momentum wave, one may write: p2/2m fp + Sum k Vk f(p-k) = E fp where W= Sum over p fp exp(ipx) in one dimension ((1)) The objective of this note is to observe differences in the interaction of a single plane wave with a potential ((1)) and the interaction of W with V in the Schrodinger equation. In particular, for a particle in a box with infinite walls, V(x)=0 inside the one dimensional box, yet the Fourier series of W includes all p values. From ((1)), even though V(x)=0 as an average, there seems to still be an interaction with various Vk values (Vk= the kth Fourier transform) in the interior of the box as ((1)) holds at all x points.
In Part II we noted that in the literature the Schrodinger equation is sometimes derived (e.g. (1)) from two fluid equations which follow from Boltzmann’s transport equation, namely the continuity equation (conservation of mass) and a conservation of momentum equation. We also suggested the following flow/flux equations for a free particle: d/dx partial A = p ((1a)) and d/dt partial A = -E ((1b)) or A=-Et+px a Lorentz invariant. For a quantum free particle one may convert these into eigenvalue equations (see Part II). For a bound state one has a system with no overall net momentum, but an overall energy. Thus one may expect the free particle quantum solution exp(-iEt) to still appear. In (1), u(x,t) = velocity field is defined as d/dx S(x,t) based on the idea of irrotational flow. We argued that for u(x,t)=0 S(x,t)=f(t) where t is any function which is ambiguous. Furthermore, in order to have the equations of (1) apply for a bound state S=-Et or in other words there is a principle that d/dt S = -E i.e ((1b)). In this note we examine both a free quantum particle and a bound state in terms of the fluid approach of (1). The result S=-Et remains, but in the case of the free particle ((1a)) emerges from dS/dx = u(x,t)=p. For the bound state overall p is 0, but S=-Et still holds. As a result there is a density from the fluid approach and a related stress potential which leads to a conservation of energy equation. The stress potential, however, is equal to the average kinetic energy at each x because E= W(x)+V(x) from (1). In (1) W(x) is calculated using assumptions linking sij = < (vi-ui)(vj-uj)> to derivatives of density. We calculate average kinetic energy directly using: P(p/x) = a(p)exp(ipx) / Wv(x) where Wv(x) =wavefunction = Sum over p a(p)exp(ipx). The two results are the same.
In classical mechanics, the spatial density d(x) of a particle moving under a conservation of energy law .5mv(x)v(x) + V(x) = E is proportional to 1/v(x). This is obtained from d(x)= dt/T, i.e. the probability to be at x equals the time spent at x divided by the cycle time T. Given that dx=vdt to first order, density is proportional to 1/v(x). Now, d(x) is supposed to match the time independent quantum mechanical density W(x)W(x), where W(x) is the wavefunction, for high energy levels. Thus, one would expect the time a p wave (plane wave) spends at a point should be part of time independent quantum mechanics, but it does not seem to be overtly present. Furthermore, there is the question of T for the quantum mechanical problem. It is known that even for a time-independent solution W(x) has a factor exp(iEt). Thus, it seems time is present, even if it disappears in the density W*(x)W(x). It is argued that W*(x)W(x) = P(x) where P is probability, but W(x) is related to conditional probabilities such as P(x/p) or P(p/x). Thus, these conditional probabilities should be related to time. In an earlier note, we tried to show the time-independent Schrodinger equation implies all p plane waves are cycling with the same frequency E. In this note, we try to find a quantum mechanical representation for P(x,p)=dt/T where dt(x,p) is the time spent at x by a plane wave p and also try to evaluate T.