Considering a Formula that Holds True inside a Hydrogen Atom that Is Derived Based on Einstein's Energy-Momentum Relationship that Holds True in Free Space
| Scientific Paper | |
|---|---|
| Title | Considering a Formula that Holds True inside a Hydrogen Atom that Is Derived Based on Einstein's Energy-Momentum Relationship that Holds True in Free Space |
| Read in full | Link to paper |
| Author(s) | Koshun Suto |
| Keywords | Einstein's Energy-Momentum Relationship, Special Theory of Relativity, Dirac Equation, Rest Mass Energy, Potential Energy. |
| Published | 2009 |
| Journal | General Science Journal |
| No. of pages | 16 |
Read the full paper here
Abstract
Einstein's energy-momentum relationship, which is representative of the Special Theory of Relativity, holds true in an isolated system in free space, but quantum mechanics as represented by the Dirac equation has been considered the best theory to describe the behavior of electrons possessing potential energy inside atoms. This paper asks the question: "if a formula similar to Einstein's relationship, which holds true in free space, were to also hold true inside a hydrogen atom, what would such a formula look like?" It then derives this relationship. The newly derived formula of this paper could prove useful as a formula to supplement quantum mechanics.
Overview
This is Koshun Suto's fullest statement of a claim he develops across several papers: that Einstein's energy–momentum relationship E2 = c2p2 + (mec2)2 does not hold for an electron bound inside a hydrogen atom, and that the formula which does hold there is Eab,n2 + c2pn2 = (mec2)2. The c2p2 term changes sign and crosses to the other side of the equation, so that as the electron's momentum increases its total energy falls below the free-space rest energy rather than rising above it.
The framing is deliberately conservative. Suto does not claim quantum mechanics is wrong — "this paper does not disagree with this theory" — and he presents the result as a supplement rather than a replacement. His motive is symmetry: physicists abandoned the energy–momentum relationship at the atomic boundary and handed the problem to the Dirac equation, but nobody asked what the analogous intra-atomic relation would look like. He argues the question should have been posed in the 1920s, while quantum mechanics was still incomplete, and that an answer developed without relying on the completed theory is therefore legitimate. The companion paper Alternative Formulation of Quantum Mechanics takes the same relation and asks what coefficients a Dirac-type equation would need in order to satisfy it; this one stops at the relation itself and its energy levels, noting that "the issues of quantizing Eq. (19) will be the topic of another paper."
The argument
Why Einstein's relation is said to fail inside the atom
The case is made by contrasting two histories for the same electron. An electron at rest in free space has only its rest mass energy E0 = mec2; if it absorbs a photon, that energy becomes kinetic and the total energy rises, so E > E0 and Einstein's relation applies. But an electron drawn toward a proton "emits a photon from itself without absorbing external energy, and at the same time gains an amount of kinetic energy equivalent to the photon energy." Its total energy therefore falls, to E < mec2, while its momentum grows. Suto stresses a bookkeeping point: the Bohr convention setting E = 0 for an electron at rest at infinity is a convention about differences, whereas the Dirac eigenvalue for hydrogen, E = mec2[1 − α2/2n2 − (α4/2n4)(n/k − 3/4)], is defined on an absolute scale that includes the rest energy. Read that way, the Bohr energy En is the reduction in the electron's rest mass energy, and mec2 + En is what remains.
A prior result of Suto's is imported at this point and does real work in the derivation: that the potential energy of a bound electron equals the reduction in its rest mass energy, Δ(mec2) = V(r), citing his paper in Physics Essays 22, 135 (2009). Appendix A supplies the supporting virial relations from the circular Bohr orbit: mev2/r = e2/4πε0r2, hence V(r) = −2(½mev2), E = −K and E = V(r)/2, with the shortfall carried off as radiation so that V(r) + K + ħω = 0.
The derivation
Suto follows the textbook derivation of the free-space relation (crediting A. P. French) but reverses one differential. Where the free case has dE = dK and therefore dE = vdp, the bound case has −dE = dK and therefore −dE = vdp. Combining p = mev with E = mc2 to give c2p = Ev, and multiplying, yields EdE = −c2pdp, which integrates to E2 = −c2p2 + const. He notes the constant "should normally [be] determined through experimentation," but assigns it the value (mec2)2 "from the analogy" with the free-space relation. Defining the absolute total energy as Eab,n = mec2 + Kn + V(rn) = mec2 + V(rn)/2 = mec2 + En gives the paper's central formula, (mec2 + En)2 + c2pn2 = (mec2)2. He raises the objection to himself that a non-relativistic expression has been inserted into a relativistic one, and answers that the free-space relation is itself routinely applied to slow particles.
Checks and consequences
Section 4 confirms that the new relation returns the familiar momenta. Classical quantum theory gives pn = (1/n)(mee2/4πε0ħ); expanding the new relation gives pn2 = (mec)2(α2/n2 − α4/4n4), and since α2 is of order 5 × 10−5 the second term is negligible, leaving pn ≈ αmec/n — the same value. Section 5 then rearranges the relation as Eab,n = [(mec2)2 − c2pn2]1/2 = mec2(1 − α2/n2)1/2 and expands binomially to mec2(1 − α2/2n2 − α4/8n4 − ⋯). This is shown to coincide exactly with the Dirac eigenvalue whenever the radial quantum number n′ = 0, since with s = (k2 − α2)1/2 and n = n′ + k one has [1 + α2/(n′ + s)2]−1/2 = (1 − α2/n2)1/2. Writing k = j + ½ and tabulating all levels for n = 1, 2, 3, the matching ones are 1S1/2, 2P3/2 and 3D5/2 — the highest level of each shell.
This paper goes one step beyond its companion. Rather than leaving the agreement confined to those levels, Suto conjectures in Eq. (49) that Eab simply is the Dirac eigenvalue in general, and writes the relation with the full j-dependent Dirac expression substituted for E: E2 + c2p2 = (mec2)2 with E = mec2[1 − α2/2n2 − (α4/2n4)(n/(j+½) − 3/4)]. The practical claim of the conclusion is one of economy: obtaining the fine-structure energies took Dirac a route through the Klein–Gordon equation and "a great deal of effort," whereas here they follow from a binomial expansion.
Appendix B supplies a supporting puzzle. Gasiorowicz's relativistic scalar equation for a bound electron corresponds to (E − V)2 = c2p2 + (mec2)2; but if E is the Bohr energy then E − V = K, which would demand K2 > (mec2)2, an inequality that "should normally not be possible." Suto's resolution is that the E there must be the absolute quantity mec2 − K, which on substitution returns Einstein's relation exactly.
Assessment
The paper's best feature is that its central mathematical observation is correct and checkable. mec2(1 − α2/n2)1/2 really does reproduce the Dirac hydrogen eigenvalue on the n′ = 0 branch, and Suto proves the identity through s = (k2 − α2)1/2 rather than asserting it; his table correctly picks out 1S1/2, 2P3/2, 3D5/2. The framing question is also a fair one to ask, and rarely asked: the change of description at the atomic boundary really is a discontinuity in physical practice rather than a derived result. The Appendix B observation about the scalar relativistic equation is a genuine inconsistency in a standard textbook presentation, and identifying which E is meant is the right diagnosis. Compared with much of the literature that attacks relativity, this paper is precise about what it claims and explicit about what it has not done.
The weaknesses are concentrated in two places. First, the constant of integration is chosen, not determined. Suto says as much, and the choice (mec2)2 is exactly what makes Section 5 agree with Dirac; the agreement is therefore weaker evidence than it appears. Second, the reversed differential −dE = vdp describes how energy and momentum vary across a family of stationary states as n changes, not how they vary for one particle in motion. Integrating it as though E and p were the dynamical variables of a single particle conflates two different things, and this is why the result is, as Suto concedes, "not a quantized expression" that can yield only degenerate-state energies. The imported premise Δ(mec2) = V(r) — that binding literally reduces the electron's rest mass — is also not argued here but taken from another paper, and it is a strong claim: the standard account attributes the mass defect of a bound system to the system as a whole, not to a change in the electron's own rest mass, and no measurement of a free electron liberated from an atom shows it returning from a reduced mass.
The extrapolation in Eq. (49) is the most exposed step. Having shown agreement only for the n′ = 0 levels, Suto asserts that Eab equals the full j-dependent Dirac eigenvalue. But his own derivation makes Eab a function of n alone — it contains no quantity that distinguishes 2S1/2 from 2P3/2 — so the substitution introduces a j dependence the formula cannot generate. That matters against measurement: the fine-structure splitting of the n = 2 levels of hydrogen, about 10.9 GHz between 2P1/2 and 2P3/2, is precisely the j dependence in question, and it is measured to many significant figures. A formula that assigns one energy per shell does not predict it. The same limitation leaves the Lamb shift between 2S1/2 and 2P1/2 — about 1.06 GHz, and a degeneracy the Dirac equation itself does not break — entirely outside the paper's reach, as it must be for any approach that stops short of field theory.
Judged against its own stated ambition, which is to supply a supplementary formula and a simpler route to the degenerate energies rather than to replace the Dirac equation, the paper largely does what it promises. Judged as a physical claim about the electron's rest mass changing inside the atom, it rests on a premise argued elsewhere and on an integration constant chosen for convenience, and it has not earned the general form asserted at the end.