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 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 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 a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
In a previous note (1) we argued that the notion of a quantum wavelength follows from the comparison of the classical energy conservation equation for an oscillator with the time-independent Schrodinger equation. In particular for the classical oscillator pp/2m + k/2 xx = E transforms into cos(wt)cos(wt) + sin(wt)sin(wt) = 1 which shows the oscillatory nature of the system explicitly. The time independent Schrodinger equation must also fit this form in the sense that kinetic energy and potential energy must become functions of a variable y such that there are no coefficients. E on the RHS is multiplied by a factor and cannot depend on parameters such as k or m and so is proportional to sqrt(k/m) i.e. the angular frequency. As a result momentum squared scales as 1/xx and momentum as 1/x. Thus momentum is of the form of 1/wavelength.These arguments seem to rely only on nonrelativistic physics. In a previous note (2) we argued that the quantum wavelength follows from p scaling x in the Lorentz invariant -Et+px. This is strictly a relativistic argument so how can it be compatible with the nonrelativistic oscillator argument suggested above? We use the result of (3) in which we argue that for a relativistic Klein-Gordon oscillator near the turning points velocity must be very low and then become 0. Thus a relativistic oscillator equation should become a nonrelativistic one in the neighbourhood of turning points. In other words the time-independent Schrodinger equation should hold in such a neighbourhood. Classically the form cos(wt)cos(wt)+sin(wt)sin(wt) should also hold. Thus p behaving as 1/x must apply both in the relativistic scenario because one has the Klein-Gordon equation, but also due to the classical oscillator scaling situation because the time-independent Schrodinger equation holds in the vicinity of the endpoints. Thus both arguments for p behaving as 1/x should apply and be consistent with each other.
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).
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.
In bound state quantum mechanics, one has statistical observables (based on P(x) where P is probability) such as spatial density d(x), potential energy density V(x)d(x) and kinetic energy density .5mv(x)v(x) d(x), where v(x) is the classical velocity obtained from .5mv(x)v(x) + V(x) = E. (Kinetic energy density can also be written as -1/2m W(x) d/dx d/dx W(x) where W(x) is the wavefunction). The energy of a state can be obtained by integrating these statistical observables i.e. Integral dx d(x) V(x) + Integral dx d(x) .5mv(x)v(x) = E. If one performs this integration for two different states, one may calculate the energy difference. This begs the question: why does quantum mechanics deal with conditional probabilities such as P(p/x) directly related to W(x)? The issue perhaps begins with the appearance of V(x) and E in the time independent Schrodinger equation. Kinetic energy (not kinetic energy density) should appear and be somehow linked to some motion. A question arises as to why the energy density is proportional to the spatial density i.e. Energy density = E d(x)? This is directly linked to the time-independent Schrodinger equation, but one may try to argue that classical energy conservation .5mv(x)v(x) + V(x) = E at each point x should hold in quantum mechanics with .5mv(x)v(x) equal to Sum over p p*p/2m P(p/x) where P(p/x) = fp exp(ipx)/W(x). If one, however, considers excited states written as Wn(x)= Wo(x)Hn(x) where Wo(x) is the ground state wavefunction, one finds that: -1/2m d/dx d/dx Hn(x) - 1/m [d/dx Wo(x)]/Wo(x) [d/dx Hn(x)] = E1 Hn(x) ((1)) In other words, an eigenvalue equation appears again with no V(x) present. If one assumes Hn(x) has physical meaning (i.e. is related to a phonon or photon), then ((1)) may suggest there is a physical principle in place, i.e. the idea that one must have a “constant” energy at each point x. Here E1 is strictly kinetic energy (although not all of it). This “principle” has consequences as it restricts the values of E1 (i.e. discrete eigenvalues) and leads often to a periodic Hn(x) which has direct effect on kinetic energy density and potential energy density. Thus, having Hn(x) interact with Wo(x) to produce a constant energy E1 times Hn(x) does affect kinetic and energy densities at each x point, just not kinetic energy itself (it is constant i.e. E1). The purpose of this note is to examine this possible principle behind quantum mechanics.