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.
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.
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 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.
If a Lagrangian contains a potential dependent on dx/dt, it is possible this does not contribute to the classical equations of motion. Nevertheless, the fact that the potential is present suggests there is a physical impact. A different Lagrangian yielding the same equations of motion, does not necessarily represent the same physical situation i.e. the equations of motion do not uniquely determine a physical situation. In particular, using the Lagrange formalism yields momentum=p = dL / d (dx/dt) which may contain the usual m dx/dt term plus terms from velocity related potentials. We argued in previous notes, that for quantum mechanics there is a momentum distribution at each x point which yields an average pp/2m at x. Moving from one x to another yields a change in pp indicating an average acceleration. The potential V(x), it was argued, may be written as Sum over k Vk exp(ikx). Thus, it is stochastic, yielding momentum hits of k which only yield a force -dV/dx on average. Given that average acceleration is stochastic, average momentum should also be stochastic. For the case of a potential A(x)dx/dt, the Lagrangian momentum dL/d (dx/dt)= mdx/dt +A(x). dx/dt is an average of stochastic value i.e. Sum over p a(p) exp(ipx), but A(x) is also stochastic i.e. A(x)= Sum over k Ak exp(ikx). One may combine the two terms to create a term of the form bk exp(i j x). We argue that this contains the combined stochasticity of p and A(x). Special considerations show that if W(x)=Wo(x)W1(x), then one may break out W1(x) as a phase related directly to A(x) i.e. W1(x)= exp{ i (Integral dx1 (0,x) A(x1) } which is experimentally observable.
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).
In Part I, we focused on the special relativistic Lorentz invariants EE = pp + momo (c=1) and (E-V)(E-V) = pp + momo (c=1). We argued that these together with the variable switching forms (familiar in thermodynamics e.g. E(S,V) but F=E-TS → F(T,V)) i.e. E = -L1(v) + pv and (E-V) = -L1(v) + pv lead to p=dL/dv (partial). From this, one may show using the variable switching forms that an equation of motion follows from the extremization of L. Here we focus instead on the Lorentz invariant -Et+px (for a free particle) which we write as: L1(v)t = t(-E+pv) and L1(v)t = t( -(E-V) + pv) for a particle with a potential. We show again that dL/dv partial = p from which the extremization of L =L1-V follows. Interestingly, this same Lorentz invariant with another partial derivative linked to p, namely Action = Lt = -Et+px → dAction/dx = p. In this case, the independent variables differ. Forming an eigenvalue equation: -id/dx exp(iA) = p exp(iA) leads to quantum mechanics. This time, in the case of a potential one cannot simply use E→ E-V(x) as many exp(ipx)s are stirred up to create the distribution W(x)=Sum over p a(p)exp(ipx). If one, however, uses -id/dx as a momentum operator (which follows from -id/dx exp(ipx) = p exp(ipx)) and applies this to the Lorentz operator: (E-V)(E-V) = pp + momo (c=1) then one obtains the Klein-Gordon equation (id/dt partial-V) (id/dt partial -V) W = -d/dxd/dx W + momo W. Thus a quadratic form holds with E-V, but the linear form -Et+px only holds for a single p in the quantum (but not the classical) case.
In classical mechanics, it is possible to have different Lagrangians yield identical equations of motion. For example, a velocity dependent potential x dx/dt does not affect the equations of motion, but leads to a different definition for momentum i.e. p= mdxd/dt - x versus p = mdx/dt. (In general, dG(x)/dt behaves in a similar manner.) In quantum mechanics, p→ -i d/dx and so mdxd/dt - x is the momentum used. Given a bound state wavefunction W(x) written as a Fourier series, it is the new momentum which is represented by p. This p which is not mdx/dt is involved in forming quantum resonances i.e. solutions of the Schrodinger equation. These resonances, however, do not necessarily affect the quantum partition function because the extra potential may create a phase, similar to the result of writing the Schrodinger equation viewed from a moving frame. Thus, one may have two different Hamiltonians physically equivalent to: m/2 dx/dt dx/dt + V(x) namely: p/2m + V(x) where p=mdx/dt and (p+x)(p+x)/2 + V(x) where p=mdx/dt - x. In this note, we examine the partition function from the Feynman path integral approach. Using this propagator approach one may find different partition functions for each case, even though E= m/2 dx/dt dx/dt + V(x). Setting X=Y, however, yields a P(x) which is the same. We consider the case of an extra potential of d/dt G(x).