In some recent articles in the literature (1),(2) the Schrodinger equation: p*p/2m W(x) + V(x) W(x) = E W(x) ((1)) is replaced with an equation using a relativistic momentum operator: [sqrt(p*p + m*m) - m] W(x) + V(x)W(x) = E W(x) ((2)) The first operator on the LHS of ((2)) is then expanded in powers of 1/(m*m). Next, p is replaced with hbar/i d/dx (or gradient in 3D) and perturbation theory used. In this note, we consider p*p as being a conditional average given by: p*p= [Integral dp f(p) exp(ipx) p*p ] / [Integral dp f(p) exp(ipx)] ((3)) The denominator is the wavefunction W(x). This result should hold for the nonrelativistic case. In the relativistic case, p is replaced everywhere with prel where prel= p/ sqrt(1-p*p/(m*m)) prel*prel = Integral dprel p*p (1+ p*p/(m*m)) f(prel) exp(iprel x) ((4)) And W(x) = Integral dprel f(prel) exp(iprel x) ((5)) To first order, we argue that Integral dprel f(prel) exp(iprel x) = Integral dp f(p) exp(i p x), but Integral p*p f(prel) dprel exp(i prel x) contributes a term . For higher orders, even Integral dprel f(prel) exp(i prel x) contributes. Thus, we try to argue that one does not only have a change in the prel*prel operator, but one must consider changes of prel within Integral dprel f(prel) exp(iprel x) itself. Thus, perturbation theory may not be enough in such cases.
In some recent articles in the literature (1),(2) the Schrodinger equation: pp/2m W(x) + V(x) W(x) = E W(x) ((1)) is replaced with an equation using a relativistic momentum operator: [sqrt(pp + mm) - m] W(x) + V(x)W(x) = E W(x) ((2)) The first operator on the LHS of ((2)) is then expanded in powers of 1/(mm). Next, p is replaced with hbar/i d/dx (or gradient in 3D) and perturbation theory used. In this note, we consider pp as being a conditional average given by: pp= [Integral dp f(p) exp(ipx) pp ] / [Integral dp f(p) exp(ipx)] ((3)) The denominator is the wavefunction W(x). This result should hold for the nonrelativistic case. In the relativistic case, p is replaced everywhere with prel where prel= p/ sqrt(1-pp/(mm)) prelprel = Integral dprel pp (1+ pp/(mm)) f(prel) exp(iprel x) ((4)) And W(x) = Integral dprel f(prel) exp(iprel x) ((5)) To first order, we argue that Integral dprel f(prel) exp(iprel x) = Integral dp f(p) exp(i p x), but Integral pp f(prel) dprel exp(i prel x) contributes a term . For higher orders, even Integral dprel f(prel) exp(i prel x) contributes. Thus, we try to argue that one does not only have a change in the prel*prel operator, but one must consider changes of prel within Integral dprel f(prel) exp(iprel x) itself. Thus, perturbation theory may not be enough in such cases.
In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.
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 (1), standard quantum perturbation theory is developed using Vo(x)+bV(x), W(x)=Wo(x)+bn1(x)+b*bn2(x)+... and E=Eo+E1b+b*bE2+... where b is small. These values are inserted into the time-independent Schrodinger equation and terms with like order of b collected. Multiplying by Wo and integrating over x yields an expression for Ei. Multiplying by Wk(x) and integrating yields a projection of <ni | k> where Wk is the kth eigenfunction of -1/2m d/dxd/dx +Vo(x). The projection <ni|Wo> is determined using <W |W>=1 and retaining terms of b to the power i. Thus, many terms are dropped from <W|W> and the approach used in (1) is a little tedious as noted in that reference. In this note, we consider W(x)=Wo(x)W1(x) which yields: -1/2m d/dx d/dx W1 + d/dx W1 d/dx Wo / Wo + b V(x) W1(x) = (E-Eo) W1(x) when inserted into the Schrodinger equation. The same <ni|Wk> projection results, as expected. For <ni|Wo>, however, we argue that <WoW1 | WoW1>=1 leads to a more direct and less tedious equation for evaluating this projection. We interpret the results which are based on eigenfunctions of Ho i.e. the Hamiltonian with bV(x) dropped. Finally, we compare the high energy quantum perturbed density with the classical (correspondence) principle i.e density = C/v(x) where v(x) is velocity. In the high energy classical limit, matters are not so simple because an expansion of the spatial density in positive powers of b is only valid away from the classical turning points. Near the turning points, a different power series in b holds and technically there is a region for which there is no power series as E-Vo and b(E1-V) are very similar in value.
In a quantum bound state,spatial density is given by W(x)W(x), where W(x) is the wavefunction, average momentum by -i d ln(W) /dx and average kinetic energy at x by -1/2m d/dx d/dx W / W. One may note that: -1/2m d/dx d/dx W / W + V(x) = E is the time independent Schrodinger equation which may be solved for W(x) and E without any concept of spatial density. Furthermore, one may calculate and <p*p>, also without any concept of spatial density. This differs from the case of a classical fluid situation for which there is no W(x). We argue that kinematics of a bound state are contained in W(x) and that the Schrodinger equation is like a probability conservation hydrodynamic equation with spatial density replaced with W(x) and a V(x)W(x) term which we try to explain. In addition, the “conservation of momentum” hydrodynamic equation is essentially “d/dx” of the Schrodinger equation (if mass=m=.5). This strategy differs from the general approach used throughout the literature (for example (1) and many other cases), where spatial density is taken as the starting point of a quantum hydrodynamic formalism. Spatial density is utilized in a continuity equation of the form d/dt [W*W] + d/dx [W*W v] = 0 where v is a collective velocity and d/dt the partial derivative with respect to time. We argue one only needs the concept of a wavefunction as the normalization constant of the conditional probability P(p/x)= a(p)exp(ipx) / W(x) i.e. W(x)= sum over p a(p) exp(ipx). From this wavefunction, one may calculate and <p*p>. Thus we argue that hydrodynamic type equations should apply to W(x). We pointed out in (2), that W(x) in fact satisfies a momentum continuity hydrodynamic equation if mass=.5.
In Newtonian mechanics, force and energy are clearly separated. For example, pp/2m = Integral F dx = Integral dp/dt dx = m/2 vv = pp/2m. The equation pp/2m + V(x) = E is considered to be strictly an energy equation. In special relativity, mocc is considered to be strictly an energy. We have suggested in previous notes that mo is proportional to inertia which is a force. Thus, mo and mocc are thought of in two very different manners simultaneouly, i.e. as an energy and as a force. We propose that it might be beneficial to think of both mo and V(x) in terms of energy and force distribution simultaneously. In particular, we consider that V(x) is not only potential energy at x, but is also linked to force or interaction at x through -d/dx V(x). The reason this duality of energy-force may be important is that it is directly linked to quantum mechanics, we argue. In general, it is argued that classical mechanics is a high energy level limit of quantum mechanics, We argue, however, that for a single particle bound state, the energy linked with a momentum p (at any energy level) is pp/2m (nonrelativistic) which is deterministic (and also follows from special relativity) and has nothing to do with quantum mechanics. The difference between classical mechanics and quantum, is that there exists an interaction (impulse hit) probability of exp(ipx) which means that instead of a hit occurring at the center-of-mass point, it may occur within an hbar/p range. Thus, one may have a particle interact and move through either of two slits if they are separated by about hbar/p. If p, however, is changed in an interaction, p1->p2, then the change in energy p2p2/2m - p1p1/2m (one dimension) is classical-deterministic. It is, however, the duality of V(x) with force that force one to consider not only energy pp/2m (deterministic), but also its associated force distribution given by exp(ipx) / W(x), where W(x) = Sum over p a(p)exp(ipx). As a result, one creates the average kinetic energy term: {Sum over p a(p) pp/2m exp(ipx) } / W(x). One may call exp(ipx)/W(x) a probability and so formally one has KEave(x), but physically exp(ipx) represents a force distribution. The balance to V(x) is a combination of a deterministic energy pp/2m with a force distribution exp(ipx)/W(x). This, we argue, is a physical statement and helps to describe why V(x) may be expanded as a series of virtual particles (e.g. photons etc). It is not just energy, but force that must be considered together. It is only in classical physics, pp/2m + V(x) = E that one solely considers the energy aspects, i.e. pp/2m and V(x) only as energy, but the more general picture, we argue, is to consider not only energy, but force distribution together. This is the quantum picture and it incorporates not only probability, but deterministic Newtonian mechanics (or special relativity) through the use of pp/2m.