Alternative Formulation of Quantum Mechanics
| Scientific Paper | |
|---|---|
| Title | Alternative Formulation of Quantum Mechanics |
| Read in full | Link to paper |
| Author(s) | Koshun Suto |
| Keywords | Einstein's energy-momentum relationship, Special Theory of Relativity, Dirac equation. |
| Published | 2008 |
| Journal | General Science Journal |
| No. of pages | 16 |
Read the full paper here
Abstract
For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein's energy-momentum relationship represents the relationship between the particle's total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein's relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.
Overview
Koshun Suto argues that Einstein's energy–momentum relationship, E2 = c2p2 + E02, has a limited domain of applicability, and that a different relation governs an electron bound inside a hydrogen atom. The claimed replacement is Eab,n2 + c2pn2 = E02 — the momentum term changes sign and moves to the other side, so that binding lowers the total energy below the rest energy while the momentum grows. The paper's second half asks what happens to the Dirac equation if this new relation, rather than the Klein–Gordon equation, is the constraint that its coefficients must satisfy.
The physical picture behind the sign change is stated plainly. In free space, adding energy to a particle raises both its kinetic energy and its total energy, so dE = dK and hence dE = vdp. But an electron falling toward a proton radiates photons out of the atom: its total energy falls while its kinetic energy rises. For that case Suto writes −dE = dK, and therefore −dE = vdp. Integrating this reversed differential is what flips the sign of the c2p2 term. Suto is careful to say the result is not meant to refute Dirac: the modified equations are offered "as alternative forms of the Dirac equation," a second representation standing to the usual one, in his analogy, as the Heisenberg picture stands to the Schrödinger picture.
The argument
Energy on an absolute scale
The paper first insists on a bookkeeping change. Classical quantum theory sets the zero of energy at an electron at rest infinitely far from the nucleus, so the Bohr levels En are negative and Kn = −En. Suto notes that the Dirac eigenvalue for hydrogen, expanded in powers of the fine structure constant α, is instead defined on an absolute scale that includes the rest energy: to order α4 it is E = mc2[1 − α2/2n2 − (α4/2n4)(n/k − 3/4)]. Dropping the third term recovers the Bohr formula added to mc2. He therefore defines the absolute total energy of a bound electron as Eab,n = E0 + Kn + V(rn) = E0 + V(rn)/2 = E0 + En, which is strictly less than E0. Appendix A supplies the virial relations used: from mv2/r = e2/4πε0r2 one gets V(r) = −2(½mv2), hence E = −K and E = V(r)/2, with the difference between potential and kinetic energy carried away as radiated ħω so that V(r) + K + ħω = 0.
Deriving the intra-atomic relation
The derivation follows the textbook route (Suto credits A. P. French) but with the reversed differential. Combining p = mv and E = mc2 gives c2p = Ev. Multiplying this by −dE = vdp yields EdE = −c2pdp, whose integral is E2 = −c2p2 + const. Suto fixes the constant as E02 "from the analogy" with the free-space case, and substitutes Eab,n for E, obtaining
- Eab,n2 + c2pn2 = E02, or (E0 + En)2 + c2pn2 = E02.
He acknowledges the awkwardness of inserting a non-relativistic expression into a relativistic equation, and defends it by noting that the free-space relation itself is normally applied to slow particles without difficulty.
Consistency check on the momentum
Section 3 verifies the new relation reproduces known momenta. From Kn = pn2/2m and the Bohr energy, classical quantum theory gives pn = (1/n)(me2/4πε0ħ). Expanding the new relation instead gives E02(1 − α2/2n2)2 − c2pn2 = E02, whence pn2 = (mc)2(α2/n2 − α4/4n4). Because α2 is of order 5 × 10−5, the second term is negligible and pn ≈ αmc/n, which is the same expression. Suto takes this agreement as showing the relation "has been shown to be true for an electron inside a hydrogen atom."
Energy levels without solving the Dirac equation
Section 4 rearranges the new relation as Eab,n = (E02 − c2pn2)1/2 = mc2(1 − α2/n2)1/2, and expands by the binomial theorem to Eab,n = mc2(1 − α2/2n2 − α4/8n4 − ⋯). This is the same as the Dirac expansion when the radial quantum number n′ = 0, that is when n = k. Suto shows the identity algebraically: with s = (k2 − α2)1/2 and n = n′ + k, the Dirac eigenvalue [1 + α2/(n′ + s)2]−1/2 reduces to (1 − α2/n2)1/2 exactly when n′ = 0. Writing k = j + ½ and tabulating, the matching levels are those with j = l + ½ at maximum l: 1S1/2, 2P3/2, 3D5/2, which he identifies as the highest level available for each n. He presents this as a labour-saving result — the fine-structure term is obtained by a binomial expansion instead of "complex calculations" — while conceding that because his expression is not quantized it delivers only these degenerate-state energies.
New coefficients for the Dirac equation
Quantizing the free-space relation by E → iħ∂/∂t, p → −iħ∇ gives the Klein–Gordon equation, and Dirac's coefficients αi, β are those that make the squared linear operator reproduce it. Quantizing Suto's relation instead flips the sign of the Laplacian term, so the required conditions on the new coefficients become α′i2 = −1 (rather than +1), with α′iα′j + α′jα′i = 0, α′iβ′ + β′α′i = 0 and β′2 = 1. He exhibits a 4 × 4 solution: α′1 and α′2 are the usual off-diagonal matrices with factors of i inserted, α′3 likewise, and β′ = β unchanged. The point of the exercise is stated in the conclusion: the standard treatment of hydrogen keeps Dirac's free-space coefficients and adds a potential term −V to the Hamiltonian, whereas here V is already inside Eab,n, so no potential term is needed and the modification is carried entirely by the coefficients.
Appendix B supplies a motivating puzzle. Gasiorowicz's relativistic scalar equation for a bound electron corresponds to (E − V)2 = c2p2 + E02; but E − V = K + V − V = K, which would require K2 > E02, "this kind of inequality should normally not be possible." Suto's resolution is that the E appearing there must be the absolute quantity E0 − K, in which case substitution returns Einstein's relation exactly.
Assessment
What is attractive here is the economy of the central algebraic observation. It is genuinely true, and not widely remarked, that mc2(1 − α2/n2)1/2 reproduces the Dirac hydrogen eigenvalue exactly on the n′ = 0 branch, and Suto proves it rather than asserting it, tracing the identity back through s = (k2 − α2)1/2. His table of levels is correct in identifying 1S1/2, 2P3/2, 3D5/2 as the matching set. The Appendix B observation is also a fair one: the scalar relativistic equation with E interpreted as the Bohr energy really is inconsistent, and asking which E is meant is the right question. And the paper is unusually modest about its own claims — it explicitly does not seek to displace the Dirac equation.
The difficulties are at the load-bearing steps, and they are of a kind the paper does not acknowledge. The constant of integration is not determined; Suto says outright that it "should normally [be] determined through experimentation" and then sets it to E02 "from the analogy" with the free-space case. But that choice is precisely what makes the result come out matching Dirac, so the agreement of Section 4 is partly built in rather than derived. Worse, the two relations are then in conflict for the same electron: the free-space relation and the intra-atomic relation share the same E0 and would both have to hold in the limit of vanishing binding, where they agree only because p → 0. The reversed differential −dE = vdp is a statement about a bound electron's trajectory through states, not about its instantaneous dynamics; integrating it as though E and p were the conjugate variables of a single free particle conflates a family of stationary states with the motion of one particle. That is why the resulting expression, as Suto concedes, "is not a quantized expression" and yields energies only for the degenerate cases.
The matrix result should also be read carefully. Requiring α′i2 = −1 is equivalent to multiplying Dirac's αi by i, which is exactly what his displayed matrices do; the operator is then anti-Hermitian in its spatial part, so the corresponding Hamiltonian is not Hermitian and the usual guarantees of real eigenvalues and conserved probability do not follow. The paper does not check either. Nor does it show that the modified equation actually reproduces the hydrogen spectrum when solved — the spectrum is obtained separately, from the binomial expansion, and the matrices are only shown to satisfy the algebraic conditions. The claim that this dispenses with the potential term is therefore not demonstrated: the coefficients are constants, so they cannot carry the r-dependence that V(r) supplies.
Against measurement, the exposed point is that the construction reproduces one level per principal quantum number and is silent about the rest. Fine structure is a splitting: the Dirac formula's dependence on j is what accounts for the observed 2P1/2–2P3/2 separation in hydrogen, measured to high precision, and Suto's expression assigns no energy at all to the levels his own table leaves unmarked. It also has nothing to say about the Lamb shift between 2S1/2 and 2P1/2, which is a degeneracy the Dirac equation itself does not break. Taken as what it says it is — an alternative representation with the potential absorbed into the definition of total energy, offered without claiming to overturn anything — the paper is a coherent exercise. Taken as a rival formulation of quantum mechanics, as the title suggests, it is far short of the required demonstration. A companion paper, 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, develops the same relation further.