The Relativistic Bohr Radius and Its Agreement with the Dirac Most-Probable Radius

1. The Bohr Radius

Bohr [1] introduced his atomic model in 1913. In the second part of his trilogy, he derived the radius now known as the Bohr radius,

$$a_0={\displaystyle \frac{4\pi {ϵ}_0{\hslash }^2}{e^2{m}_e}}.$$

(1)

Using the fine-structure constant, this can be rewritten as

$$a_0={\displaystyle \frac{\hslash }{m_{e}c\alpha }}={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}\approx 5.29\times {10}^{-11}\mathrm{m},$$

(2)

where ${ϵ}_0=1/\left({\mu }_0{c}^2\right)$ is the permittivity of free space, $\alpha$ is the fine-structure constant, $m_e$ is the electron rest mass, e is the elementary charge, $\hslash =h/\left(2\pi \right)$ is the reduced Planck constant, and ${\overline{\lambda }}_e=\hslash /\left(m_{e}c\right)$ is the reduced Compton wavelength of the electron. The Compton wavelength itself was not introduced until 1923 by Arthur H. Compton [2], a decade after Bohr’s original papers.

When the finite proton mass is included, the reduced-mass-corrected Bohr radius for the hydrogen atom is

$$a_0^{*}={\displaystyle \frac{m_e}{\mu }}{a}_0={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}\left(1+{\displaystyle \frac{{\overline{\lambda }}_P}{{\overline{\lambda }}_e}}\right)={\displaystyle \frac{{\overline{\lambda }}_e+{\overline{\lambda }}_P}{\alpha }},$$

(3)

where $\mu =m_e{m}_P/(m_e+m_P)$ is the reduced mass and ${\overline{\lambda }}_P$ is the reduced Compton wavelength of the proton. According to NIST CODATA, the Bohr radius is

$$a_0=5.29177210544\left(82\right)\times {10}^{-11}\mathrm{m}.$$

Its uncertainty is linked mainly to the uncertainties in the fine-structure constant and the electron reduced Compton wavelength.

There are several equivalent ways to obtain the Bohr radius. One standard route equates the Coulomb force with the centripetal force,

$$\begin{array}{cc}\hfill k_e{\displaystyle \frac{e^2}{r^2}}& =m_e{\displaystyle \frac{v^2}{r}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\hslash c\alpha }{r^2}}& =m_e{\displaystyle \frac{v^2}{r}}.\hfill \end{array}$$

(5)

For the ground state in the Bohr model, $v=\alpha c$, and therefore

$$\begin{array}{cc}\hfill r& ={\displaystyle \frac{\hslash c\alpha }{m_e{c}^2{\alpha }^2}}={\displaystyle \frac{\hslash }{m_{e}c\alpha }}={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}.\hfill \end{array}$$

(6)

Another derivation uses de Broglie’s standing-wave condition. If the electron circumference is an integer multiple of the de Broglie wavelength ${\lambda }_b=h/\left(mv\right)$, then

$$2\pi r=n{\displaystyle \frac{h}{mv}}.$$

(7)

Equivalently, the angular momentum is quantized:

$$mvr=n\hslash .$$

(8)

Solving for v gives

$$v={\displaystyle \frac{n\hslash }{m_{e}r}}.$$

(9)

Substituting this expression into the force-balance condition gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{m_e}{r}}{\left({\displaystyle \frac{n\hslash }{m_{e}r}}\right)}^2& ={\displaystyle \frac{\hslash c\alpha }{r^2}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill r_n& =n^2{\displaystyle \frac{\hslash }{m_{e}c\alpha }}=n^2{\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}.\hfill \end{array}$$

(11)

Thus the first Bohr radius is obtained for $n=1$.

The Schrödinger equation confirms the Bohr radius as the most probable radius for the electron in the hydrogen ground state. However, the Schrödinger equation is non-relativistic. This leaves open the question of how the Bohr-radius scale should be adjusted when relativistic momentum is included.

The main result of the present paper is that Bohr’s own relativistic prescription leads to precisely the same radius as the most-probable radius of the Dirac $1s_{1/2}$ radial probability density for a point-nucleus, one-electron Coulomb system. This turns the relativistic Bohr radius from a purely historical curiosity into a compact bridge between the semi-classical Bohr model and the relativistic Dirac description.

2. The Relativistic Bohr Radius and the Relativistic Compton Wavelength

Bohr was aware that his stated radius formula rested on non-relativistic assumptions. In Part II of his trilogy, he indicated that the formula could be extended to cases in which the orbital velocity is significant relative to the speed of light by replacing the mass term with

$${\displaystyle \frac{m}{\sqrt{1-\frac{v^2}{c^2}}}}=m\gamma ,$$

(12)

where

$$\gamma ={\displaystyle \frac{1}{\sqrt{1-v^2/c^2}}}$$

(13)

is the Lorentz factor.

It is important to distinguish this historical prescription from later debates over “relativistic mass.” The expression $m\gamma$ is naturally understood here through relativistic momentum,

$$p=m\gamma v,$$

(14)

which was introduced by Planck [4]. The use of $m\gamma$ in the orbital momentum does not require treating relativistic mass as an ontologically separate mass concept; it is enough to use the standard relativistic momentum.

Following Bohr’s stated prescription, the relativistically adjusted Bohr radius becomes

$$a_{0,r}={\displaystyle \frac{4\pi {ϵ}_0{\hslash }^2}{e^2{m}_e\gamma }}.$$

(15)

This can be written as

$$a_{0,r}={\displaystyle \frac{\hslash }{m_e\gamma c\alpha }}={\displaystyle \frac{{\overline{\lambda }}_e\sqrt{1-\frac{v^2}{c^2}}}{\alpha }}.$$

(16)

The numerator is the Lorentz-contracted reduced Compton wavelength of the electron. For the hydrogen ground state in the Bohr model, $v=\alpha c$, and therefore

$$a_{0,r}={\displaystyle \frac{{\overline{\lambda }}_e\sqrt{1-{\alpha }^2}}{\alpha }}\approx 5.29149\times {10}^{-11}\mathrm{m}.$$

(17)

The difference between the non-relativistic and relativistic expressions is

$$\begin{array}{cc}\hfill a_0-a_{0,r}& ={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}-{\displaystyle \frac{{\overline{\lambda }}_e\sqrt{1-{\alpha }^2}}{\alpha }}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }}\left(1-\sqrt{1-{\alpha }^2}\right)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \approx {\displaystyle \frac{{\overline{\lambda }}_e\alpha }{2}}.\hfill \end{array}$$

(20)

Numerically, this is approximately

$$a_0-a_{0,r}\approx 1.41\times {10}^{-15}\mathrm{m}.$$

The relative correction is

$${\displaystyle \frac{a_0-a_{0,r}}{a_0}}=1-\sqrt{1-{\alpha }^2}\approx {\displaystyle \frac{{\alpha }^2}{2}}\approx 2.66\times {10}^{-5},$$

(21)

or about $0.00266\%$.

The same expression follows from using the relativistic de Broglie relation,

$${\lambda }_b={\displaystyle \frac{h}{m\gamma v}},$$

(22)

rather than the non-relativistic approximation ${\lambda }_b=h/\left(mv\right)$. The angular-momentum quantization condition becomes

$$m_e\gamma vr=n\hslash .$$

(23)

Solving for v gives

$$v={\displaystyle \frac{n\hslash }{m_e\gamma r}}.$$

(24)

Substitution into the force-balance condition yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{m_e\gamma }{r}}{\left({\displaystyle \frac{n\hslash }{m_e\gamma r}}\right)}^2& ={\displaystyle \frac{\hslash c\alpha }{r^2}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill r_n& =n^2{\displaystyle \frac{\hslash }{m_e\gamma c\alpha }}=n^2{\displaystyle \frac{{\overline{\lambda }}_e\sqrt{1-\frac{v^2}{c^2}}}{\alpha }}.\hfill \end{array}$$

(26)

For $n=1$, this reduces to Eq. (16).

The factor

$${\overline{\lambda }}_{e,r}={\overline{\lambda }}_e\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}={\displaystyle \frac{\hslash }{m_e\gamma c}}$$

(27)

is identical in form to the relativistically adjusted reduced Compton wavelength recently discussed by Haug. In ordinary Compton scattering, Compton [2] considered the scattering of photons by electrons initially treated as stationary. A relativistic Compton wavelength instead takes into account the electron’s motion at the time of interaction. The ordinary Compton wavelength is

$$\lambda ={\displaystyle \frac{h}{mc}},$$

(28)

whereas the corresponding relativistic expression is

$${\lambda }_r={\displaystyle \frac{h}{m\gamma c}}.$$

(29)

3. Relation to Relativistic Rydberg Formulae

The traditional Rydberg formula is also non-relativistic in origin. It has often been used to infer the electron mass and, indirectly, the electron Compton wavelength. Bohr relied heavily on the Rydberg formula in developing his atomic model, but his explicit relativistic correction to the formula appears only in his 15 April 1914 letter to Fowler.

In modern notation, the standard Rydberg relation for hydrogenic transitions can be written as

$${\displaystyle \frac{1}{\lambda }}=R\left({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}}\right),$$

(30)

where R is the appropriate Rydberg constant, with reduced-mass corrections included when necessary. Bohr’s early relativistic attempt introduced a correction term of the general form

$$\nu ={\displaystyle \frac{2{\pi }^{2}m{e}^4{N}^2}{{\hslash }^3}}{\displaystyle \frac{M}{M+m}}\left({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}}\right)\left[1+{\displaystyle \frac{{\pi }^2{e}^4{N}^2}{{\hslash }^2{c}^2}}\left({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}}\right)\right],$$

(31)

where the notation follows the older form used by Bohr. This correction did not provide satisfactory agreement with the observed spectral data.

More recent work has proposed relativistically adjusted Rydberg formulae [8]. Such formulae have also been discussed in connection with high-n Rydberg transition spectroscopy in high-energy plasma contexts, including W7-X stellarator studies [9,10]. In such systems, fully quantum-electrodynamic calculations can become highly demanding, especially for large atoms and high-energy environments. Relativistically corrected semi-classical expressions may therefore be useful as approximate tools, provided their assumptions and limitations are made explicit.

The main point of the present paper is not that the Bohr model replaces modern relativistic quantum theory. Rather, the point is that Bohr’s own suggested relativistic adjustment leads directly to a length scale of the form

$${\displaystyle \frac{{\overline{\lambda }}_{e,r}}{\alpha }},$$

where ${\overline{\lambda }}_{e,r}$ is a relativistically contracted reduced Compton wavelength. This link clarifies why the Bohr radius, the Compton wavelength, and the Rydberg constant are not independent conceptual objects, but closely connected expressions of the same underlying constants.

4. Comparison with Dirac-equation Radius Concepts

The decisive question is whether the semi-classical radius in Eq. (16) corresponds to any well-defined radius concept in relativistic quantum mechanics. Since the Dirac equation provides the standard relativistic one-electron description of hydrogen-like atoms, the appropriate comparison is with radius measures derived from the Dirac wavefunction.

There is not a single unique “Dirac radius.” Several different quantities can be defined. The expectation value ${\langle r\rangle }_D$ measures the mean distance from the nucleus. The most-probable radius, by contrast, is the radius at which the radial probability density is maximal. These two quantities are already different in the non-relativistic Schrödinger theory: for the hydrogen ground state the most-probable radius is $a_0$, whereas the expectation value is $3a_0/2$.

The result below shows that Bohr’s relativistic prescription singles out the Dirac most-probable radius, not the Dirac expectation value. This is the central observation of the paper. For a point nucleus, infinite nuclear mass, and a one-electron Coulomb field, the replacement $\alpha \to Z\alpha$ gives

$$a_{0,r}^{\left(Z\right)}={\displaystyle \frac{a_0}{Z}}\sqrt{1-Z^2{\alpha }^2},$$

(32)

and this is exactly the maximum of the Dirac $1s_{1/2}$ radial probability density. Existing work on Dirac-based corrections to the Bohr radius, such as Buitrago [14], provides useful reference formulas for expectation and maximum-probability distances. The additional point emphasized here is the explicit identification of Bohr’s historical relativistic prescription with the Dirac most-probable radius.

4.1. Numerical Comparison for Hydrogen-Like Ground States

A useful numerical comparison can be made for hydrogen-like one-electron ions in the Dirac $1s_{1/2}$ ground state. This is the cleanest case because there is only one electron and no electronic screening or shell-structure effects. The comparison should therefore not be confused with empirical atomic radii of neutral atoms, which can vary non-monotonically with Z because of shell filling and many-electron interactions.

For a point nucleus and infinite nuclear mass, the Dirac radial probability density for the $1s_{1/2}$ state may be written, up to a normalization constant, as

$$P_D\left(r\right)\propto r^{2{\gamma }_Z}exp\left(-{\displaystyle \frac{2Zr}{a_0}}\right),{\gamma }_Z=\sqrt{1-Z^2{\alpha }^2}.$$

(33)

The maximum of this radial probability density is therefore obtained from

$${\displaystyle \frac{d}{dr}}\left[r^{2{\gamma }_Z}exp\left(-{\displaystyle \frac{2Zr}{a_0}}\right)\right]=0,$$

(34)

which gives

$$r_{\mathrm{mp}}^D={\displaystyle \frac{a_0}{Z}}{\gamma }_Z={\displaystyle \frac{a_0}{Z}}\sqrt{1-Z^2{\alpha }^2}.$$

(35)

This is the same functional form as the semi-classical relativistic Bohr radius obtained by replacing $\alpha$ with $Z\alpha$ in Eq. (16):

$$a_{0,r}^{\left(Z\right)}={\displaystyle \frac{a_0}{Z}}\sqrt{1-Z^2{\alpha }^2}.$$

(36)

Thus, within the assumptions of a point nucleus, infinite nuclear mass, and a one-electron Coulomb field, the semi-classical relativistic Bohr radius is exactly identical to the Dirac most-probable radius for the $1s_{1/2}$ radial probability distribution. This equality is the main result of the paper.

The Dirac expectation value of the radius is different. From the same radial probability density one obtains

$${\langle r\rangle }_D={\displaystyle \frac{a_0}{Z}}{\displaystyle \frac{2{\gamma }_Z+1}{2}}.$$

(37)

This reduces to the familiar non-relativistic Schrödinger value $\langle r\rangle =3a_0/\left(2Z\right)$ in the limit $Z\alpha \ll 1$, but it is not identical to the most-probable radius. The distinction is important: Eq. (36) agrees with the Dirac most-probable radius in Eq. (35), but it should not be identified with the Dirac expectation value in Eq. (37).

Using the 2022 CODATA values $\alpha =7.2973525643\times {10}^{-3}$, $a_0=5.29177210544\times {10}^{-11}\mathrm{m}$, and ${\overline{\lambda }}_e=3.8615926744\times {10}^{-13}\mathrm{m}$ [13], the hydrogen ground-state result is

$${\gamma }_1=\sqrt{1-{\alpha }^2}=0.9999733739683034,$$

(38)

$$a_{0,r}=a_0{\gamma }_1=5.29163120655\times {10}^{-11}\mathrm{m},$$

(39)

so that

$$a_0-a_{0,r}=1.40898891811\times {10}^{-15}\mathrm{m}.$$

(40)

The relative correction is

$${\displaystyle \frac{a_0-a_{0,r}}{a_0}}=2.66260317\times {10}^{-5},$$

(41)

or approximately $0.0026626\%$. The Dirac expectation value for the same state is

$${\langle r\rangle }_D={\displaystyle \frac{a_0}{2}}(2{\gamma }_1+1)=7.93751725927\times {10}^{-11}\mathrm{m},$$

(42)

whereas the non-relativistic Schrödinger value is $3a_0/2=7.93765815816\times {10}^{-11}\mathrm{m}$.

4.2. Why the Schr ödinger Expectation Value Is Larger

The non-relativistic Schrödinger value $3a_0/\left(2Z\right)$ is correct, but it is not the quantity that should be compared directly with the Bohr radius. The Bohr radius corresponds to the maximum of the radial probability distribution for the hydrogenic $1s$ state, not to the mean radius. For the Schrödinger $1s$ wavefunction, the radial probability density is

$$P_S\left(r\right)\propto r^{2}exp\left(-{\displaystyle \frac{2Zr}{a_0}}\right).$$

(43)

Maximizing this expression gives

$$r_{\mathrm{mp}}^S={\displaystyle \frac{a_0}{Z}},$$

(44)

which is the usual Bohr-radius result for a hydrogen-like ion. However, the expectation value is obtained by averaging over the whole radial distribution,

$${\langle r\rangle }_S={\displaystyle \frac{{\int }_0^{\infty }r{P}_S\left(r\right)dr}{{\int }_0^{\infty }{P}_S\left(r\right)dr}}={\displaystyle \frac{3a_0}{2Z}}.$$

(45)

It is therefore larger because the radial probability distribution has a long tail toward larger radii. The peak of the distribution occurs at $a_0/Z$, but the average is pulled outward by this tail. This is not a relativistic effect; it already occurs in the ordinary non-relativistic hydrogen ground state.

The same distinction remains in the Dirac theory. The relativistic Bohr radius derived in this paper agrees with the Dirac most-probable radius $r_{\mathrm{mp}}^D$, not with the Dirac mean radius ${\langle r\rangle }_D$. Thus, the fact that $3a_0/\left(2Z\right)$ or ${\langle r\rangle }_D$ is larger than $a_{0,r}^{\left(Z\right)}$ is not a contradiction. It simply reflects the difference between the location of the maximum probability and the average position of the electron.

The numerical sequence in Table 1 is the direct comparison relevant to the present paper. It should decrease smoothly with Z. If a table of “Dirac radii” appears to jump up and down as one moves to higher elements, then it is almost certainly mixing different physical quantities, such as neutral-atom radii, screened many-electron radii, empirical covalent or van der Waals radii, or radii belonging to different orbitals.

The expectation value ${\langle r\rangle }_D$ is a different radius concept and should not be placed in the same column as the Bohr-radius comparison. For the same idealized $1s_{1/2}$ Dirac-Coulomb state it is

$${\langle r\rangle }_D={\displaystyle \frac{a_0}{Z}}{\displaystyle \frac{2{\gamma }_Z+1}{2}}.$$

(46)

This quantity is larger than the most-probable radius, just as in the non-relativistic Schrödinger hydrogen ground state where $\langle r\rangle =3a_0/2$ while the most-probable radius is $a_0$. Therefore ${\langle r\rangle }_D$ is useful as a separate Dirac benchmark, but it is not the radius that coincides with Eq. (16).

The high-Z entries in Table 1 should nevertheless be interpreted as idealized point-nucleus Dirac-Coulomb estimates. For large Z, finite nuclear size, reduced-mass effects, QED corrections, and nuclear-structure effects become increasingly important. In addition, the simple expression above applies only while $Z\alpha <1$ and only to hydrogen-like one-electron ions. It should not be extrapolated directly to neutral atoms or to multi-electron ions without including screening, electron-electron interaction, and shell structure.

5. Discussion

The correction derived above is small for ordinary hydrogen, but it is not conceptually negligible. For hydrogen, the correction to the most-probable radius is approximately $1.41\times {10}^{-15}\mathrm{m}$, corresponding to about $2.66\times {10}^{-5}$ of the ordinary Bohr radius, or $0.0026626\%$. This length is comparable to nuclear length scales, even though it is tiny compared with the Bohr radius itself.

The key result is stronger than a mere numerical similarity. The semi-classical expression obtained by following Bohr’s relativistic prescription has the same analytic form as the maximum of the Dirac $1s_{1/2}$ radial probability density:

$$a_{0,r}^{\left(Z\right)}=r_{\mathrm{mp}}^D={\displaystyle \frac{a_0}{Z}}\sqrt{1-Z^2{\alpha }^2}.$$

(47)

This gives a precise interpretation to the relativistic Bohr radius: in the ideal one-electron Dirac-Coulomb problem, it is the most-probable radius, not the expectation value. The distinction matters because ${\langle r\rangle }_D$ is larger than $r_{\mathrm{mp}}^D$, just as in the non-relativistic hydrogen ground state the mean radius $3a_0/2$ is larger than the most-probable radius $a_0$. The larger expectation value should therefore not be interpreted as a failure of the Bohr-radius comparison; it answers a different question, namely the average radius of the probability distribution rather than the radius at which the probability density is maximal.

The numerical values for hydrogen-like ions should not be confused with atomic radii of real neutral elements. In the hydrogen-like one-electron sequence, the directly comparable radius $a_{0,r}^{\left(Z\right)}=r_{\mathrm{mp}}^D$ varies smoothly with Z and decreases as Z increases. Non-monotonic behavior, such as radii that jump up and down across the periodic table, is expected only when one includes many-electron effects, shell filling, electronic screening, changes of orbital, or empirical radius definitions. These are different physical quantities from the hydrogen-like Dirac radii considered here.

Several qualifications are essential. First, the Bohr model remains semi-classical and cannot reproduce the full spectrum, spin structure, or full wavefunction content of relativistic hydrogen. Second, the Dirac equation already provides the standard relativistic quantum-mechanical treatment of hydrogen-like atoms. Third, the exact equality found here applies specifically to the maximum of the $1s_{1/2}$ radial probability density under idealized point-nucleus and infinite-nuclear-mass assumptions. For high-Z ions, finite nuclear size, reduced-mass effects, QED corrections, and nuclear-structure effects should be included before comparing with precision spectroscopic data.

Subject to these qualifications, the agreement is noteworthy. It shows that Bohr’s neglected relativistic instruction does not merely produce a plausible contraction of the Bohr radius; it reproduces the radius selected by the relativistic quantum-mechanical probability maximum. This makes the result a useful historical and conceptual bridge between Bohr’s model, the reduced Compton wavelength, and the Dirac-Coulomb theory of hydrogen-like ions.

6. Conclusions

We have revisited Bohr’s little-discussed relativistic prescription for the Bohr radius. The ordinary Bohr radius,

$$a_0={\displaystyle \frac{4\pi {ϵ}_0{\hslash }^2}{e^2{m}_e}}={\displaystyle \frac{\hslash }{m_{e}c\alpha }}={\displaystyle \frac{{\overline{\lambda }}_e}{\alpha }},$$

(48)

is based on a non-relativistic momentum scale. If the relativistic momentum factor is included, the ground-state radius becomes

$$a_{0,r}={\displaystyle \frac{{\overline{\lambda }}_e\sqrt{1-{\alpha }^2}}{\alpha }}=a_0\sqrt{1-{\alpha }^2}.$$

(49)

For hydrogen-like one-electron ions this generalizes to

$$a_{0,r}^{\left(Z\right)}={\displaystyle \frac{a_0}{Z}}\sqrt{1-Z^2{\alpha }^2}.$$

(50)

The main result of the paper is that this expression is exactly equal to the most-probable radius obtained from the Dirac $1s_{1/2}$ radial probability density, under the assumptions of a point nucleus, infinite nuclear mass, and a one-electron Coulomb field:

$$a_{0,r}^{\left(Z\right)}=r_{\mathrm{mp}}^D.$$

(51)

It is not equal to the Dirac expectation value ${\langle r\rangle }_D$, which is a different radius measure. This distinction resolves possible confusion about which Dirac radius is being compared with the relativistic Bohr radius. The same distinction is already present in the non-relativistic Schrödinger theory, where the most-probable radius is $a_0/Z$ but the expectation value is $3a_0/\left(2Z\right)$ because the radial probability distribution has an extended tail toward larger radii.

For hydrogen, the correction is

$$a_0-a_{0,r}={\displaystyle \frac{{\overline{\lambda }}_e(1-\sqrt{1-{\alpha }^2})}{\alpha }}\approx {\displaystyle \frac{{\overline{\lambda }}_e\alpha }{2}}\approx 1.41\times {10}^{-15}\mathrm{m},$$

(52)

corresponding to approximately $0.0026626\%$ of the ordinary Bohr radius. Although small, the correction is conceptually significant because it identifies a direct link between Bohr’s semi-classical relativistic prescription and the Dirac-Coulomb probability maximum.

To our knowledge, this is the first explicit demonstration that Bohr’s historical relativistic radius prescription yields the Dirac most-probable radius for the hydrogen-like $1s_{1/2}$ state. The result should be understood within its idealized assumptions and should not be generalized directly to neutral atoms, many-electron systems, empirical atomic radii, or high-Z precision spectroscopy without including reduced-mass, finite-nuclear-size, QED, and many-body effects.