In a previous note (1) we examined the idea of a classical frequency such as that of an oscillator appearing in quantum mechanical results i.e. solutions for the quantum oscillator. In this note, we examine the problem differently. We begin with the quantum oscillator time-independent Schrodinger equation and note that the LHS becomes an equation in the variable y only where yy = sqrt(km) xx with E → E/w. This is already known in the literature, but it suggests E/w must be independent of k and m. For example, if the LHS only has the variable y with no k or m present and y is integrated from -infinite to infinite, E/w should also have no k, m dependence. At this point, however, w is not really a frequency in the quantum sense. It is a frequency in the classical analysis so one should compare the quantum and classical cases. One may consider the classical conservation of energy equation and integrate over x which may be converted into an integral over t. “wt” then allows one to integrate over a cycle, but E/w appears on the RHS as in the quantum case. This seems to show that quantum mechanics picks out special classical energies for which the classical cycle integral produces a value which is not dependent on k or m.
In a previous note (1) we examined the idea of a classical frequency such as that of an oscillator appearing in quantum mechanical results i.e. solutions for the quantum oscillator. In this note, we examine the problem differently. We begin with the quantum oscillator time-independent Schrodinger equation and note that the LHS becomes an equation in the variable y only where yy = sqrt(km) xx with E → E/w. This is already known in the literature, but it suggests E/w must be independent of k and m. For example, if the LHS only has the variable y with no k or m present and y is integrated from -infinite to infinite, E/w should also have no k, m dependence. At this point, however, w is not really a frequency in the quantum sense. It is a frequency in the classical analysis so one should compare the quantum and classical cases. One may consider the classical conservation of energy equation and integrate over x which may be converted into an integral over t. “wt” then allows one to integrate over a cycle, but E/w appears on the RHS as in the quantum case. This seems to show that quantum mechanics picks out special classical energies for which the classical cycle integral produces a value which is not dependent on k or m.
In Parts I and II we noted that a quantum oscillator time-independent Schrodinger equation may be scaled such that x→y with yy=sqrt(km)xx and E→ Esqrt(m/k). Given that the LHS side is a function strictly of y which may be integrated from -infinite to infinite, the RHS must be independent of k and m. Thus E is proportional to sqrt(k/m) which “ happens” to be the oscillator classical angular frequency. In Part II we noted that the scaling occurs in the Schrodinger equation because one uses -id/dx as the momentum operator which follows from the exp(ipx) form of a free particle wavefunction. In this note, we show that for any equation: ad/dx d/dx W + b xx W = E W scaling of the form x→y such that yy =xx sqrt(b/a) and E→ E 1/sqrt(ba) occurs. Furthermore if d/dx is associated with momentum then writing conservation of energy using m/2 vv leads to 1/a www cos(wt)cos(wt) + b sin(wt)sin(wt) = b and w is proportional to sqrt(ab). Thus E must be proportional to w, the classical frequency. (An equation of the form a df/dt df/dt + b f f = constant must have a cos(wt) solution as is already known.) We try to extend these ideas by considering a wave on a string with both ends fastened. The string produces sinusoidal waves, but sound is created if there is air. One may consider sound to come in identical units of energy proportional to the angular frequency w. This is similar to the Planck blackbody radiation assumption about light. It may be shown that the string follows an oscillator solution because d/dx dy/dx which represents the change in the sin(theta) multiplied by the constant tension from x to x+dx is set equal to y i.e Force is proportional to y which is the oscillator result. If one sets energy proportional to w as in the case of light and wishes to have an equation strictly in x describing this scenario and also linked to energy conservation then the momentum squared term should have dimensions 1/xx. This suggests d/dx d/dx as a solution because E must be proportional to w. Thus we argue that one may work “backwards” modeling phonons like photons in a classical problem to obtain the quantum operator p = -id/dx which is linked to the free particle wavefunction exp(ipx).
In Part III we argued that a Schrodinger equation of the form: a d/dx d/dx W(x) + bxx W(x) = E W(x) may be scaled x→y such that yy = xx sqrt(ba) and E→ E sqrt(a/b) so that E is proportional to sqrt(b/a). We applied this idea to the quantum oscillator and noted that classically w, the angular frequency, for the same equation is sqrt(b/a). We further argued that this implies the momentum operator in quantum mechanics should scale as 1/xx as d/dx d/dx does. In this note we use a slightly different argument to analyze this same problem. Instead of treating the classical oscillator and time-independent Schrodinger equation as separate, we link them and show that scaling must occur with E being proportional to w. Furthermore we argue that if quantum mechanically one wishes to use probabilities f(p,x) to calculate average kinetic energy at x i.e. KE(at x) = {Sum over p pp/2m f(p,x) } / {Sum over p f(p,x)}, then f(p,x) = f(px) and pp should scale as 1/xx. Thus comparison of the classical oscillator equation directly with a statistical average type of quantum equation which uses V(x), E and KE(at x) yields information about E, f(p,x) and p i.e. introduces the idea of wavelength. Furthermore the discrete quantum energy values E=hbar w (n+.5) are dependent on w not through any boundary value considerations.
In Part I we noted that the quantum oscillator result E= hbar w (n+.5) implies that for exp(-iEt) there are special times proportional to 1/E which in turn are proportional to 1/w which is proportional to the classical cycling time of the oscillator. In other words for oscillators with different w there are different 1/w times such that -Et (which is part of a free particle action A= -Et+px) is constant i.e. one cannot distinguish as in statistical mechanical equilibrium. In this note, we consider the free particle action in more detail. From A=-Et+px there are special times (1/E) and lengths 1/p proportional to wavelength such that different E and p values when multiplied by these times and length have the same value. Furthermore A=0 for this values. We argue that it is this feature of px in exp(ipx) which is responsible for the energy of an oscillator being proportional to w, the classical angular frequency. Thus the idea of a special length (wavelength) such that different p’s when multiplied by 1/p produce the same value leads to a quantum energy such that 1/E is proportional to a physical cycling time in the classical problem. Thus different w’s lead to E/E being constant just as p/p is constant.
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 (1), a method for finding the propagator of a generalized quantum oscillator directly from the time-dependent Schrodinger equation is presented. This method assumes the form Q(t)exp[i (B(t)xx + A(t)yy + C(t)xy) ] which is inserted into the Schrodinger equation. The equation is broken into purely imaginary and real parts. Time dependent coefficients of xx, xy and yy are found leading to four differential equations which may be solved to ultimately yield a full solution for the propagator. In (2), it is suggested that the propagator is given by Q(t) exp(i Integral (t=0 to Tf) Lc dt1)). In order to use this approach, a solution to the classical Lagrangian Lc is found and x written as: XA(Tf-t)+YB(t-0) where X and Y are the initial and final x positions (and are numbers). Thus, dx/dt may be computed, and the classical action integral (t=0 to Tf) Lc dt) found through integration. In this note, we consider the quantum oscillator and try to demonstrate the two approaches are the same. A key point is that -id/dt (partial) in the Schrodinger equation really represents -i d/dTf (partial).
The time-independent Schrodinger equation: En= -1/2m [d/dx dWn/dx] / Wn(x) + V(x) ((1)) matches a classical conservation of energy equation where -1/2m [d/dx dWn/dx] / Wn(x) = KEclassical(x). In classical physics, however, one has x(t) so each x point is associated with a specific t for a given energy En. In the quantum time-independent equation, one has exp(-iEnt) as the time dependent factor and does not consider any link with x although if one is comparing ((1)) to a classical conservation equation then a link between x and t could be made to complete the analogy. This is usually not considered, but we examine this point as we argue it becomes important in the time-dependent situation. For the time-dependent Schrodinger case we consider two approaches. First, a linear combination of time independent solutions is taken i.e. W(x,t) = Sum over n an(En) exp(-iEnt) Wn(x) ((2)) Again each Wn(x) has an association with KEclassical(x) and so one may think in terms of (x,t) classical pairs for a given En. The second approach is to consider the function: exp(-iEnt + m Integral (0,x) v(y) dy) ((3)). This satisfies the time-dependent Schrodinger equation and yields a classical conservation of energy equation with v(x) being the classical velocity. Noting that En= .5m v(x)v(x) + V(x), ((3)) may be rewritten as: exp(i Integral L dt) where L = T-V is the Langrangian with T=.5mv(x)v(x). At this point En has been removed from the picture, but is replaced with: x(t)= X g(t)/g(T) ((4)) where (X,T) are a space-time pair corresponding to a particular energy E value. Using exp(i Action) ((5)) in the time-dependent Schrodinger equation yields a conservation equation which holds for a particular classical energy as long as (x,t) values in the equation are classically linked to the appropriate classical (X,T) values i.e. for a classical trajectory there is a specific family of X at time T for a given energy. For another energy, there is another family of (X,T) pairs. Thus the equation obtained represents energy conservation for many different energy values depending on which simultaneous values one uses for x and t. There is, however, no overlap as a specific (X,T) family corresponds to an energy E and another nonoverlapping family of (X,T) to another energy. This, however, is the same situation as ((2)) for which there are also families of (X,T) corresponding to each En. The only difference is that the En are quantized i.e. discrete. ((3)) and ((5)) are both solutions to the time-dependent Schrodinger equation which represent a variety of energies and families of (X,T) associated to the classical solution of the energy in question. We thus suggest that ((2)) and ((4)) are equivalent for this reason without using the idea of a complete set of basis functions represented by Wn(x). We consider these ideas as the Schrodinger propagator is given by ((5)), but with ((4)) generalized to: x(t)= Xi g(tf-t)/g(tf-ti) + Xf g(t-ti)/g(tf-ti) for the case that g(0)=0 (2) so that Xi,Xf, tf and ti enter the picture. This then should be equivalent to: Sum over n exp(-iEntf) Wn(xf) Wn* (xi) ((6)) In this note, we suggest that one may link ((5)) and ((6)) based on ideas of space-time families associated with different energies as well as the two being solutions of the time-dependent Schrodinger equation.