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).
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).
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 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 classical Lagrangian L leads to Newton’s second law which is equivalent to an energy-momentum conservation equation. The classical action: A = Integral dt1 L(t1) for a constant velocity v=X/T may be varied with respect X and T i.e dA/dT = -E and dA/dX= p. Thus, one may create a differential equation in (dA/dT)(dA/dt) and dA/dx dA/dx for the EE=pp+momo equation. Alternatively, one may create other differential equations. Quantum mechanical differential equations, however, are quite different with exp(i Action) being a solution. Even this may be written as the solution of a non-Schrodinger equation i.e. id/dt exp(iAction) = -1/2m (d/dx exp(iAction)) (d/dx exp(iAction)) for a free particle. For a constant velocity, exp(iAction) is equivalent to exp(ipx-iEt). We argue that this solution is statistical because it separates x and t (as independent variables) and holds for all x and t. Thus, quantum mechanics seems to be a statistical approach using free solutions exp(ipx-iEt) to create an ensemble as in classical statistical mechanics. A difference, however, is that exp(ipx-iEt) “interferes” even if one only takes the real part cos(px-Et). We consider the case of a particle in a box with infinite potential walls to argue that the periodicity is warranted i.e. positive and negative values (interference). Even though the modulus is 1, exp(ipx-iEt) may be operated on by various generators e.g. d/dx the generator of translations. This is analogous to operations on vectors in linear algebra. As noted above, quantum equations are not only energy-momentum conservation equations. They yield a wavefunction W(x) which is an unusual momentum distribution W(x)=Sum over p a(p) exp(ipx). exp(-iEt) holds overall because average E is the same at each x. Thus, one needs a W(x) which leads to a momentum-energy conservation equation, but also satisfies certain boundary conditions which become linked to spatial density= W*W. This leads to discrete energy levels and also cases of humps and troughs in W(x). We attempt to investigate some of these ideas in this note.
In previous notes (1) we argued that the free particle action A (either nonrelativistic or relativistic) may have v (velocity) written as x/t and x and t varied independently to yield: dA/dx partial = p and dA/t partial = -E with A=-Et+px, the Lorentz invariant classical Action = t Lagrangian. We further argued that one may write eigenvalue equations i.e. -idexp(iA)/dx partial = p exp(iA) and idexp(iA)/dt partial = Eexp(iA) and call exp(iA) a type of probability. We noted that if a potential is present one may use P(p/x)= a(p)exp(ipx)/W(x) [W(x)=wavefunction = Sum over p a(p)exp(ipx)] and establish a conservation of energy equation KE(x) + V(x) = E such that there exists the same constant E at each x. We described this in two ways. First E represents binding energy so at each x one has the same binding energy i.e. each x point maps to the same E as part of the mass of an overall moving system. Secondly, we argued that E was a parameter indicating that equilibrium existed at each x point, much like temperature T. In this note we wish to consider a different idea suggested in (2), namely that if one has a quantum state: W(x)= Sum over n a(n,t) Wn(x) this may be written as: <i |H| j> a*(i,t)a(j,t) = <W H W>.. If Hamilton’s equations for complex variables are applied to these then: i d/dt partial a*(i,t) = d/da(i,t) partial { <W H W> } ((A)) (2) argues this is the time-dependent Schrodinger equation. In other words, probability weights of the Schrodinger equations (which are complex) obey Hamilton’s classical equation. In this note, we point out that the formalism dA/dx partial = p already introduces operators into classical physics, so it follows that H is an operator which may act on states. We consider the idea of ((A)) as applied to a free quantum particle moving. Next, we apply ((A)) to a quantum bound state and argue that a classical Hamilton’s equation applied to d/dt a(p) partial may describe equilibrium. Finally ((A)) may be applied to a sum of eigenstates, but this is already shown in (2).
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 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).