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.
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.
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).
In (1) the high energy limit of the Wigner transform W1(p,x,n) is investigated for the quantum oscillator. The WKB semiclassical approximation is used and the high energy wavefunction approximated by: 1/ p(x) { exp(i p(x) x) + exp(-i p(x) x) } leading to the result of delta ( PP/2m+kXX/2 - E) where p(x)= sqrt(2m(E-.5kxx)), P,X are classical momentum and position and PP/2m +kXX/2 energy. In this note, we first observe that the WKB approximation may be used for any potential V(x) to find a similar result. We then focus on obtaining the result without using the WKB approximation, but instead utilize: H(p,x) ** W1(p,x,n) = En W1(p,x,n) (from (2)) where H(p,x) is the classical Hamiltonian, ** the Moyal star product involving exp(ihbar/2 Operator) and W(p,x,n) the Wigner function (as used in (2)). The main idea is that in the classical limit, hbar—>0 so exp(ihbar/2 Operator) —>1. Thus H(p,x)W1(p,x,n→large) = En W1n(p,x,n -> large). In (2) it is shown that Integral dp dx W1(p,x,n) = 1, so Integral H(p,x) W1(p,x,n->large) = E. As a result, W1(p,x,n->large) acts as a delta function enforcing pp/2m + V(x)=E.
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.
The Maxwell-Boltzmann probability factor exp(-energy/T) / normalization may be applied to a quantum oscillator using energy levels given by hbar w (n+.5). This leads to the derivative in 1/T of a geometric series and yields an average energy: Eave = hbar w/2 + hbar w / (-1+exp(hbar w/T)). If one were to consider a semiclassical approach using p (momentum and x), energy would be described by pp/2 + xx/2 where for simplicity m=1 and k=1. This would appear in the MB factor with a temperature T and this expression would be expected to hold for high T or w/T small i.e. the quantum phonon jumps would be hardly noticeable compared to the average energy T in this limit. If one tries to write the classical energy, for w/T small, in the form of quantum energy, one has hbar w n in this limit, where n is the number of phonons (hbarw/2 is small in this limit) so: zz= {pp/2+kk/2}/w is the number of phonons which should be multiplied by hbar w/T. One may then ask: What function has the limit hbar w/T? A solution is: {1-exp(-hbarw/T)}. Thus the semiclassical normalized phase space probability: P(p,x) = {1-exp(-hbarw/T)}. Exp{ - zz {1-exp(-hbarw/T)} } yields the small hbar w/T limit quantum average energy. As a result, the average energy in this limit should match for both the classical phase space distribution and the quantum average energy expression. This classical distribution happens to be identical to the Husimi distribution. One may note (as is done in (1)), that {1-exp(-hbarw/T)} is the probability for the quantum oscillator particle to be in the ground state. In the limit of hbar w/T small, this probability is hbar w/T which is a measure of the effect of the quantum phonon leaps as well. Fisher’s information, as noted in (1), is equivalent to this ground state probability, but it is also the average energy which is proportional to the temperature type value, in this case {1-exp(-hbarw/T)}. This follows from the same kind of calculation done for a gas with particles with energy pp/2m, although instead of a factor of .5T one T because there is a contribution from pp/2 and xx/2.
In a series of previous notes (i.e. Parts II and III) we considered three methods for computing the propagator for the problem: id/dt W = -1/2m d/dx d/dx W + k(t)xxW + f(t)x (-id/dx) W. The first method is based on postulating a form Q(t) exp(-i{ b(t)XX + g(t)YY +g2(t)XY}), inserting into the time-dependent Schrodinger equation and equating coefficients of XX etc The other two methods are based on classical Lagrangians. We argued in Parts II and III that for f(t)=0 there is agreement between the methods, but issues arise with the presence of f(t)x (-id/dx) W. In this note, we try to resolve this issue by using H= p dx/dt - L ((1)) where L is the Lagrangian. In method 2, a Lagrangian with f(t)x dx/dt is considered, but this term is converted to d/dt (.5xx f) - .5xx df/dt. Thus, the time-dependent oscillator potential is modified and the d/dt [] piece thrown away because it does not affect the classical equations of motion. Method 3 (3) computes the propagator using Q(t)exp(i Action), so d/dt [] is included and leads to a phase. By using ((1)) we argue that if f(t)x dx/dt exists in the Lagrangian, one must consider a corresponding Hamiltonian of the form: H= [p+f(t)x)][p+f(t)x)] (1/2m) + k(t)xx. This is equivalent to: pp/2m + f(t)f(t)xx (1/2m) + k(t)xx + f(t)(1/m)xp -i f/2m. Thus, we see an extra f(t)f(t)xx/2m in the Hamiltonian in addition to the f(t)xp type term. This “fixes” the problem of “”ff”” terms not canceling from the Schrodinger equation in parts II and III. Thus, one must be careful when dealing with velocity dependent potential, but also note that p=mdx/dt -f(t)x now so p as in the electromagnetic case is not simply the momentum of the particle.
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.