Relativistic correction to the Bohr radius and electron distance expectation value via Dirac Equation
aa r X i v : . [ phy s i c s . g e n - p h ] O c t Relativistic correction to the Bohr radius andelectron distance expectation value via DiracEquation
October 15, 2020
J. BuitragoUniversity of La Laguna, Faculty of Physics. 38205, La Laguna (Tenerife)Spain. [email protected], [email protected]
Abstract
In this article and beginning with the Dirac solution to the Hydro-gen atom in its ground state, the exact results corresponding to theexpectation value of the distance of the electron to the proton and themaximum probability distance are found. For hydrogen-like atomsand in contrast to the non relativistic Bohr radius and expectationvalues, for Z = √
32 1 α (close to 118), the expectation distance to thenucleus is zero thus putting an end to the periodic table for Z = 118nicely matching the last element discovered (Oganesson). keywords: Dirac Equation, relativistic quantum mechanics, Bohr radius
From its appearance in 1927, the Dirac Equation has been one of the more,if not the most, studied equation in modern physics. One of the more illu-minating aspects is the treatment of the Hydrogen Atom which gives quitea few relativistic features not appearing in the paradigmatic study found in1R quantum mechanics. It comes as a surprise that until now and as faras I know, nobody has cared to obtain such a fundamental expression asthe relativistic analogue of the Bohr radius not even the expectation valuefor the distance to the nucleus in the ground state. As both results (to myknowledge never mentioned in any graduate text book) are of conceptual,pedagogical and practical value, specially for the students that find in thiscase perhaps the only one in which former fundamental ideas such as theprobability interpretation of the wave function can be carried over to therelativistic arena. Analogies with the NR treatment go even to the curiouscase in which, as in the old Bohr study of the hydrogen atom, the exact resultfor the binding energy of the ground state can be found by the same kindof simple minded approach found in the Bohr classical study (see [1]). It isthe subject of this short communication to fill these curious gaps and obtainanalytically both the expectation value and the relativistic expression of theBohr Radius thus alleviating the new comer student the feeling of havingto deal with an esoteric abstruse theory with effects such as positive andnegative energy states or the puzzling Zitterbewegung [3] (to number only afew). Indeed I consider that what the reader can be found below should beincluded in future text books dealing with the Dirac Equation.
In natural units (¯ h = c = 1), the normalized spin-up wave function solutionof the Dirac equation for the Hydrogen atom in its ground state is [2] ψ ( r, θ, ϕ ) = (2 mα ) / √ π s γ γ ) (2 mαr ) γ − e − mαr i (1 − γ ) α cos θ i (1 − γα sin θ iϕ , (1)with γ = √ − α . Γ is the Gamma function of argument (1 + 2 γ ), m the mass of the electronand α the fine structure constant. From (1), and its complex conjugate, it is2traightforward to see that the distribution of probability is isotropic: ψ ∗ ψ ∝ " − γ ) α cos θ + (1 − γ ) α sin θ . (2)The next step is to find the distance to the nucleus where the probabilityof finding the electron is maximal and the expectation value. From theprevious expression and (1): ψ ∗ ψ = 14 π m α γ ) (2 mαr ) e − mαr . (3)The expectation value < r > = Z ∞ ψ ∗ rψ πr dr, (4)follows by making 2 mαr = x and solving the integral Z ∞ e − x x γ +1 dx = Γ(2 γ + 2) , (5)together with the recursion formulaΓ(2 γ + 2) = (2 γ + 1)Γ(2 γ + 1) . The calculations can be carried out without much difficulty with theresult: < ψ | r | ψ > = 1 m s α − α . (6)In order to find the radial distance of the electron to the nucleus where theprobability of finding the electron is maximum, we consider first: ψ ∗ ψr = 2 m α π (2 mα ) γ − Γ (1 + 2 γ ) r γ e − mαr . (7)As Γ(1 + 2 γ ) is a magnitude depending only on γ , the radius of maximumprobability comes out after solving for r the equation resulting from: ddr [ r γ .e − mαr ] = 0 . r max = 1 m s α − . (8)In the non relativistic approximation equations (6) and (8) reproduce thefamiliar results of NR quantum mechanics.The last expressions are also valid for Hydrogen-like atoms making thesubstitution: α −→ Zα.
For Z = 1 /α , which is about 137, r max = 0.If the former estimation seems somewhat crude it is possible to obtain abetter sensible result. With the same substitution for α and equating to zerothe right hand side of (6) for the expectation value, we find solving for Z : Z = √
32 1 α , which is close to 118.The last, officially recognized in 2016, element is Oganesson (Og) withZ=118.