The fine structure of Weber’s hydrogen atom: Bohr–Sommerfeld approach

Abstract

In this paper, we determine in second order in the fine structure constant the energy levels of Weber’s Hamiltonian that admit a quantized torus. Our formula coincides with the formula obtained by Wesley using the Schrödinger equation for Weber’s Hamiltonian. We follow the historical approach of Sommerfeld. This shows that Sommerfeld could have discussed the fine structure of the hydrogen atom using Weber’s electrodynamics if he had been aware of the at-his-time-already-forgotten theory of Wilhelm Weber (1804–1891).

Appendices

Weber’s Lagrangian and delayed potentials

In several works, Neumann treated the connection between Weber’s dynamics and delayed potentials, see [28, 29] and [30, Ch. 8]. Neumann explained how Weber’s potential function is related to Hamilton’s principle which in [29] he called “norma suprema et sacrosancta, nullis exceptionibus obvia”.

Strictly speaking, a delay potential only makes sense for loops and not for chords. Hence we abbreviate by ${\mathcal {L}}=C^\infty \big (S^1, {\mathbb {R}}^2_\times \big )$ the free loop space on the punctured plane ${\mathbb {R}}^2_\times :={\mathbb {R}}^2 {\setminus } \{0\}$. For a potential $V \in C^\infty \big ({\mathbb {R}}^2_\times ,{\mathbb {R}}\big )$ and a constant $c _{\mathrm{W}}>0$, we define three functions

$$\begin{aligned} {\mathcal {S}}_{\mathrm {kin}}, {\mathcal {S}}_{\mathrm {pot}}, {\mathcal {S}}:{\mathcal {L}} \rightarrow {\mathbb {R}} \end{aligned}$$

by ${\mathcal {S}}={\mathcal {S}}_{\mathrm {kin}}-{\mathcal {S}}_{\mathrm {pot}}$ and by

$$\begin{aligned} {\mathcal {S}}_{\mathrm {kin}}(q):=\frac{1}{2}\int \limits _0^1|q'(t)|^2 \mathrm{d}t,\quad {\mathcal {S}}_{\mathrm {pot}}(q):=\int \limits _0^1 V\Big (q\Big (t-\tfrac{|q(t)|}{c _{\mathrm{W}}}\Big )\Big )\mathrm{d}t. \end{aligned}$$

Physically this means that the potential energy is evaluated at a retarded time. Namely, the position of the proton at the origin has to be transmitted to the electron at speed $c_{\mathrm{W}}$. It is a strange fact that to obtain Weber’s force this transmission velocity is given by the Weber constant $c_{\mathrm{W}}$ which, as measured by Weber [40], equals $\sqrt{2} c$ where c is the speed of light.

We assume that V only depends on the radial coordinate $V(q)=V(|q|)=V(r)$. Setting $r(t):=\mathopen |q(t)\mathclose |$ the functional ${\mathcal {S}}_{\mathrm {pot}}$ becomes a function of $r=r(t)$, still denoted by

$$\begin{aligned} {\mathcal {S}}_{\mathrm {pot}}(r)=\int \limits _0^1 V\Big (r\Big (t-\tfrac{r(t)}{c_\mathrm{W}}\Big )\Big )\mathrm{d}t. \end{aligned}$$

We further abbreviate

$$\begin{aligned} a=\frac{1}{c_{\mathrm{W}}}=\frac{\alpha }{\sqrt{2}},\quad c_{\mathrm{W}}=\sqrt{2} c, \end{aligned}$$

where $\alpha$ is the fine structure constant.

Setting $V_r( a,t):=V\big (r\big (t- a r(t)\big )\big )$, we obtain for the partial derivatives in a the formulas

$$\begin{aligned} \frac{\partial }{\partial a}V_r( a,t)=-V'\big (r\big (t- a r(t)\big )\big )r'\big (t- a r(t)\big )r(t) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{\partial ^2}{\partial a^2}V_r( a,t)&=V''\big (r\big (t- a r(t)\big )\big )r'\big (t- a r(t)\big )^2r(t)^2\\&\quad +V'\big (r\big (t- a r(t)\big )\big )r''\big (t- a r(t)\big )r(t)^2. \end{aligned} \end{aligned}$$

In particular, at $a=0$ we get

$$\begin{aligned} \frac{\partial }{\partial a}V_r(0,t)=-V'(r(t))r'(t)r(t) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2}{\partial a^2}V_r(0,t)=V''(r(t))r'(t)^2r(t)^2+ V'(r(t))r''(t)r(t)^2. \end{aligned}$$

We define

$$\begin{aligned} {\mathcal {S}}^k_{\mathrm {pot}}(r):=\frac{1}{k!}\int \limits _0^1 \frac{\partial ^k}{\partial a^k} V_r(0,t)\, \mathrm{d}t,\quad k\in {\mathbb {N}}_0. \end{aligned}$$

For $k=0$ this is the unretarded action functional

$$\begin{aligned} {\mathcal {S}}^0_{\mathrm {pot}}(r)=\int \limits _0^1 V(r(t))\, \mathrm{d}t. \end{aligned}$$

To see that ${\mathcal {S}}^1_{\mathrm {pot}}\equiv 0$ vanishes identically choose a primitive F of the function in one variable $r\mapsto V'(r) r$, that is $F'(r)=V'(r) r$. Indeed, we get that

$$\begin{aligned} \begin{aligned} {\mathcal {S}}^1_{\mathrm {pot}}(r)&=-\int \limits _0^1 V'(r(t))r'(t)r(t)\, \mathrm{d}t\\&=-\int \limits _0^1 F'(r(t))r'(t)\, \mathrm{d}t\\&=-\int \limits _0^1 \frac{\mathrm{d}}{\mathrm{d}t}F(r(t))\, \mathrm{d}t\\&=0. \end{aligned} \end{aligned}$$

The last equation follows, because $r(t)=\mathopen |q(t)\mathclose |$ is periodic. Using integration by parts for the second term in the sum we get that

$$\begin{aligned} {\mathcal {S}}^2_{\mathrm {pot}}(r)= & {} \frac{1}{2}\int \limits _0^1 \Big (V''(r(t))r'(t)^2r(t)^2+ \underbrace{r''(t)}_{u'}\underbrace{V'(r(t))r(t)^2}_{v}\Big )\mathrm{d}t\\= & {} \frac{1}{2}\int \limits _0^1 V''(r(t))r'(t)^2r(t)^2\,\mathrm{d}t\\&-\frac{1}{2}\int \limits _0^1 \underbrace{r'(t)}_{u}\underbrace{ \left( V''(r(t))r'(t)r(t)^2+2V'(r(t))r'(t) r(t)\right) }_{v'}\mathrm{d}t\\= & {} -\int \limits _0^1 V'(r(t))r'(t)^2 r(t) \, \mathrm{d}t. \end{aligned}$$

For the Coulomb potential

$$\begin{aligned} V(r)=-\frac{1}{r},\quad V'(r)=\frac{1}{r^2}, \end{aligned}$$

this simplifies to

$$\begin{aligned} {\mathcal {S}}^2_{\mathrm {pot}}(r)=-\int \limits _0^1 \frac{r'(t)^2}{r(t)}\,\mathrm{d}t. \end{aligned}$$

Hence Taylor approximation of ${\mathcal {S}}_{\mathrm {pot}}$ to second order in $a=\frac{1}{c_{\mathrm{W}}}$ leads to

$$\begin{aligned} \begin{aligned} {\mathcal {S}}^0_{\mathrm {pot}}(r)+\frac{1}{c _\mathrm{W}}{\mathcal {S}}^1_{\mathrm {pot}}(r)+\frac{1}{c_\mathrm{W}^2}{\mathcal {S}}^2_{\mathrm {pot}}(r)&=-\int \limits _0^1 \frac{1}{r(t)}\bigg (1+\frac{r'(t)^2}{c _\mathrm{W}^2}\bigg ) \mathrm{d}t\\&=\int \limits _0^1 S(r(t))\,\mathrm{d}t \end{aligned} \end{aligned}$$

where

$$\begin{aligned} S(r)=-\frac{1}{r}\bigg (1+\frac{r'^2}{2c^2}\bigg ) \end{aligned}$$

is Neumann’s potential function, see (2.1), and $2c^2=c _{\mathrm{W}}^2$.

Proton-proton system—Lorentzian metric

For a positive charge influenced by the proton, the Weber force exhibits fascinating properties as well. For simplicity, suppose both charges are protons and set their mass equal to one. In this case, the Lagrangian function is given by

$$\begin{aligned} L_\mathrm{W}(r,\phi ,v_r,v_\phi ) =\frac{1}{2}( v_r^2+r^2 v_\phi ^2) {-}\frac{1}{r}\bigg (1+\frac{ v_r^2}{2c^2}\bigg ). \end{aligned}$$

Changing brackets

$$\begin{aligned} \begin{aligned} L_\mathrm{W}(r,\phi ,v_r,v_\phi )&=\frac{1}{2}\bigg (1 {-}\frac{1}{c^2r}\bigg ) v_r^2 +\frac{1}{2} r^2 v_\phi ^2 {-}\frac{1}{r}\\&=\frac{1}{2}\left( g_{rr}\, v_r^2 +g_{\phi \phi }\, v_\phi ^2\right) -\frac{1}{r}. \end{aligned} \end{aligned}$$

Now the metric gets singular at Weber’s critical radius

$$\begin{aligned} \rho :=\frac{1}{c^2}=\alpha ^2. \end{aligned}$$

Outside Weber’s critical radius, the metric is Riemannian, while inside it is Lorentzian. An interesting aspect of Weber’s critical radius is that while outside Weber’s critical radius the force is repulsing—inside it is attracting! This led Weber to predict—40 years before Rutherford’s experiments—an atom consisting of a nucleus built of particles of the same charge together with particles of the opposite charge moving around the nucleus like planets. For more informations about Weber’s planetary model of the atom see [44, 45].