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 (1), standard quantum perturbation theory is developed using Vo(x)+bV(x), W(x)=Wo(x)+bn1(x)+bbn2(x)+... and E=Eo+E1b+bbE2+... 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 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 a previous note (1), we argued that W(x+dx)= W(x) exp(i dx ) where = - i d/dx lnW mimicking the behaviour of exp(ipx) for a free quantum particle. In this note, we consider that for the conditional probability P(p/x)= a(p) exp(ipx)/W(x), a shift from x to x+dx leads to: P(p/ x+dx) = P(p/x) exp(i dx (p- ) ) = P(p/x)(1+ idx (p- )). We note that in shifting from x to x+dx the overall sum of probability should not change, so P(p / x+dx)= P(p/x) + P(p/x) g(p,x). If one assumes that g(p,x) is a function of p alone, the simplest solution is p- . If one considers Sum over p pp/2m P(p/x+dx) = Sum over p pp/2m P(p/x) = delta KE ave then this should be equal to Force(x) dx. We also consider classical hydrodynamical equations and compare these with modified equations applying to the bound state quantum situation. The two sets of equations are very similar with density in the classical case being replaced with W(x)exp(-iEt), the wavefunction, and E energy or V(x) (potential) terms replaced with 2E or 2V (i.e. also 2Force). In a previous note, we had considered hydrodynamical equations with the restriction 2m=1 which is too strong. Here any m (mass) may be used.
Perturbation theory, in the arena of quantum mechanics, refers to a set of physically motivated and intuitive approximations engineered by mathematical perturbation to describe and explore the properties of a complicated system. The philosophy of perturbation theory is to begin with a simpler system, whose mathematical solution is known to the best of one’s knowledge, and then “perturb” the system by means of an externally tunable influence. For a disturbance that is not too strong, the physical quantities associated with the perturbed system can be expressed in terms of the “corrections” of the corresponding quantities of the simpler (unperturbed) system. For small perturbation, the magnitudes of these correction terms are really small in light of the magnitude of the quantities themselves and are therefore subject to treatments by approximations like the asymptotic series. The accuracy of the evaluated physical quantity, among other factors, depends on the extent to which such correction terms are incorporated. Therefore, perturbation theory is a useful avenue to explore the properties of a complicated system, where the exact (analytic) solution of the system is either difficult or impossible to achieve.
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 previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.
In the course of preparing lectures on elementary quantum theory the author developed a way of presenting time-independent perturbation theory that seems to have some advantages over more familiar presentations. There is nothing fundamentally new in the method, and if any claim of greater power can be made for the new version, it is at best for situations in which high-order perturbations are of interest. Quantum-field theorists will no doubt recognize the relation of this presentation to much that has been done in their subject recently.