The ground state wavefunction of the quantum oscillator is Wo=Cexp(-bxx) and the first excited state is W1= axWo where a, C and b are constants. Geometrically, Wo is a Gaussian centered at x=0 and x from xWo creates an orthogonal function to Wo in a simple manner by splitting a symmetric hump into two humps one positive, one negative. Now x= (dWo/dx / Wo) so W1 = dWo/dx, but d/dx is the translation generator i.e. Wo(x+dx) = Wo(x)+ dx dWo/dx. Thus not only is Wo a solution of the quantum oscillator, but its shifted change is also a solution i.e. the first excited state W1= dWo/dx. Classically translating a mass at 0 in an oscillator leads to a periodic solution. The fact that x= (dWo/dx/Wo), i.e. is a finite polynomial, has important consequences on the energy gaps of an oscillator being constant. Using Wn(x) = Wo(x)Fn(x) where Wn and Wo are the nth and o level wavefunctions in the time independent Schrodinger equation yields: -2mD(n) = d/dx dFn/dx / Fn +2 {dWo/dx/Wo}{dFn/dx/ Fn} ((1)) where En = Eo + D(n). Given that dWo/dx / Wo = Cx and noting that W1 = xWo i.e. a finite polynomial multiplies Wo one may consider Fn as a finite polynomial with a leading term of ax(to the power of n). Thus ((1)) implies that for the leading x term in any Fn(x): 2mD = 4b n where D(n) is the energy gap between the n and n+1 levels and b=.5sqt(km) where k is the oscillator constant and m the mass. Thus each level is equally spaced in energy. Thus it seems that behaviour of the oscillator ground state under translations leads to constant energy gaps. For example such behaviour does not occur for Fn(x) which is an infinite power series of x such as sin(px) which is a solution of a particle in a box with infinite potential walls.
The ground state wavefunction of the quantum oscillator is Wo=Cexp(-bxx) and the first excited state is W1= axWo where a, C and b are constants. Geometrically, Wo is a Gaussian centered at x=0 and x from xWo creates an orthogonal function to Wo in a simple manner by splitting a symmetric hump into two humps one positive, one negative. Now x= (dWo/dx / Wo) so W1 = dWo/dx, but d/dx is the translation generator i.e. Wo(x+dx) = Wo(x)+ dx dWo/dx. Thus not only is Wo a solution of the quantum oscillator, but its shifted change is also a solution i.e. the first excited state W1= dWo/dx. Classically translating a mass at 0 in an oscillator leads to a periodic solution. The fact that x= (dWo/dx/Wo), i.e. is a finite polynomial, has important consequences on the energy gaps of an oscillator being constant. Using Wn(x) = Wo(x)Fn(x) where Wn and Wo are the nth and o level wavefunctions in the time independent Schrodinger equation yields: -2mD(n) = d/dx dFn/dx / Fn +2 {dWo/dx/Wo}{dFn/dx/ Fn} ((1)) where En = Eo + D(n). Given that dWo/dx / Wo = Cx and noting that W1 = xWo i.e. a finite polynomial multiplies Wo one may consider Fn as a finite polynomial with a leading term of ax(to the power of n). Thus ((1)) implies that for the leading x term in any Fn(x): 2mD = 4b n where D(n) is the energy gap between the n and n+1 levels and b=.5sqt(km) where k is the oscillator constant and m the mass. Thus each level is equally spaced in energy. Thus it seems that behaviour of the oscillator ground state under translations leads to constant energy gaps. For example such behaviour does not occur for Fn(x) which is an infinite power series of x such as sin(px) which is a solution of a particle in a box with infinite potential walls.
The quantum oscillator ground state energy may be obtained by solving the time-independent Schrodinger equation and selecting the lowest energy eigenvalue. In (1) it is suggested that ground state oscillator energy appears in an semiclassical calculation of oscillator energy using the Moyal product (**) i.e. (p+iwx) ** (p-iwx) .5 = pp/2 + wwxx/2 + hbar w/2 for m=1. In this note, we try to show that the Moyal product: (p+iwx)/sqrt(2){ d/dx (left)d/dp (right) - d/dp (left) d/dx (right) } (p-iw)/sqrt(2) follows from the fact that Wo(x) and ao(p) where Wo is the ground state wavefunction and ao the ground state momentum wavefunction have the same functional form and that this form is in fact a Gaussian so that Wo(x)ao(p) = exp(-constant( pp/2 + wwxx/2). By using both, one may create a combined time-independent Schrodinger equation with both p and x present at the same time giving the appearance of a semiclassical approach. With respect to the ground state energy, d/dx d/dx acting on a Gaussian exp(-axx-bpp) first brings down 2ax followed by 2a from the second derivative (and similar results for d/dp d/dp). The other part of the second derivative is x dependent and cancels the x dependent potential. Thus one may think in terms of a single derivative d/dx acting on ax or a single derivative acting on bp. Combining in a symmetric form leads to the Moyal product acting on p+iwx and p-iwx to yield the ground state oscillator energy. We argue that this is essentially the same information as the time-independent Schrodinger equation with the special Gaussian form of the ground state wavefunction. Thus derivatives essentially act on pp/2 + wwxx/2 in a product of the x-wavefunction and p-wavefunction and this may be converted into a scheme of single derivative acting on the variable p+iwx and p-iwx because the relative piece of the d/dx d/dx of the Schrodinger equation is: d/dx d/dx exp(-axx) →d/dx 2ax exp(-axx) → 2a (linear in a) with the other term of the second derivative canceling with .5kxx exp(-axx).
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 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).
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.
In (1), it is argued that if the classical energy of an oscillator E = pp/2 + wwxx/2 (m=1) is written as E = (p+iwx)(p-iwx) and this in turn written using the Moyal star product { d/dp left d/dx right - d/dx left d/dp right } in terms of a parameter ( i hbar /2), one obtains: E = pp/2 + wwxx/2 + hbar w/2 i.e. a semiclassical expression for the quantum result with the ground state energy hbar w/2. In Part I, we showed that this result follows from combining a time-independent Schrodinger equation in x-space with one in p (momentum) - space. The two second order equations yield second order derivatives composed of two parts. One part returns the form pp/2m and k/2 xx and the other returns a constant which is the ground state energy if one considers the ground state p and x wavefunctions which are of the form exp(- (pp/2 + wwxx/2) /b) where b is the ground state energy.. The first part cancels with kinetic energy from the p-space equation and .5kxx from the x-space equation. Thus, this combination of two Schrodinger equations really involves three pieces: ((A)) an energy term in p,x, ((B)) a cancellation term of this energy and ((C)) a constant ground state energy term. In (2) it is stated that the Moyal star product ** has the property: Integral dx dp f(p,x) g(p,x) = Integral dx dp f(p,x) ** g(p,x) This is then used to argue that: H(p,x) ** f(p,x) = E f(p,x) ((1)). For the ground state oscillator case, only the first and second order operator terms contribute. The first order operator term is zero, while the second order term behaves in exactly the same manner as the combined x-space and p-space time-independent Schrodinger equations used in Part I. Thus the combined p and x space time-independent Schrodinger equations for the oscillator Hamiltonian pp/2 +wwxx/2 is identical to the Moyal approach.