A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?

Question

I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:

The Rydberg formula for hydrogen is written in Wikipedia as $n_2 > n_1$:

$$\frac{1}{\lambda_{\mathrm{vac}}} = R_\text{H}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$

This might be rewritten as (for any natural numbers $n_1,n_2$): $$\frac{1}{R_\text{H}\lambda_{\mathrm{vac}}} = \sqrt{\left | \frac{1}{n_1^2}-\frac{1}{n_2^2} \right |}^2 = \sqrt{\left ( \min(\frac{1}{n_1},\frac{1}{n_1})^2+\min(\frac{1}{n_2},\frac{1}{n_2})^2 -2\min(\frac{1}{n_1},\frac{1}{n_2})^2\right )}^2 = d(n_1,n_2)^2$$

or equivalently:

$$\sqrt{\frac{1}{R_\text{H} \lambda_{\mathrm{vac}}}} = d(n_1,n_2)$$

where we have set:

$$k(a,b) = \min(1/a^2,1/b^2) = \min(1/a,1/b)^2 = 1/\max(a,b)^2$$

which is a positive definite kernel on the natural numbers.

and $d(a,b) = \sqrt{k(a,a)+k(b,b)-2k(a,b)}$ is the distance measured between the two natural numbers in the Hilbert space.

Q1) Looking at the Gram matrix $G_n = (k(a,b))_{1 \le a,b \le n}$ can it be proved or disproved if the Gramian matrix has a converging spectral norm:

$$\lim_{n \rightarrow \infty} |G_n|_2 \approx 1.110288561$$

Output of the Sagemath script:

1 1.0
2 1.0756939094329987
3 1.095314794377045
4 1.1025233075784278
5 1.1057634181208507

70 1.1102858222735263
71 1.1102859354083545
72 1.1102860424258174
73 1.1102861437338427
74 1.110286239708146
75 1.1102863306951618

Q2) What is, if any, the physics interpretation of this?

The trace of $G_n$ converges to $\zeta(2)=\frac{\pi^2}{6}$, because:

$$\operatorname{tr}(G_n) = \sum_{i=1}^n k(i,i) = \sum_{i=1}^n \frac{1}{\max(i,i)^2} = \sum_{i=1}^n \frac{1}{i^2} $$

So for $n\rightarrow \infty$ we get $\zeta(2)=\frac{\pi^2}{6}$.

One possible application for physics (Q3 ?) is the following formula:

$$\pi = \arccos \left ( \frac{\frac{1}{\lambda_{xy}}+\frac{1}{\lambda_{yz}}-\frac{1}{\lambda_{xz}}}{\frac{2}{\sqrt{\lambda_{xy} \lambda_{yz}}}}\right ) +\arccos \left ( \frac{\frac{1}{\lambda_{zx}}+\frac{1}{\lambda_{xy}}-\frac{1}{\lambda_{zy}}}{\frac{2}{\sqrt{\lambda_{zx} \lambda_{xy}}}}\right ) +\arccos \left ( \frac{\frac{1}{\lambda_{yz}}+\frac{1}{\lambda_{zx}}-\frac{1}{\lambda_{yx}}}{\frac{2}{\sqrt{\lambda_{yz} \lambda_{zx}}}} \right ) $$

Every $3$ point metric space can be embedded in $\mathbb{R}^2$ as a triangle, (See https://math.stackexchange.com/questions/3393140/whats-the-name-of-this-surface-a2b2c22abc-1-0 ), and hence for each triple of distinct natural numbers $x,y,z$ we get a triangle.

For three points $x,y,z$ in a metric space, we can define (using the law of cosines) the following quantity:

$$S(x,y,z) = \frac{d(x,y)^2+d(y,z)^2-d(x,z)^2}{2d(x,y)d(y,z)}$$

We then have:

$$\pi = \arccos(S(x,y,z))+\arccos(S(z,x,y))+\arccos(S(y,z,x))$$

In the case of the Rydberg formula, it follows with $\frac{1}{\lambda_{ab} R_H} = d(a,b)^2$ and

$$S(x,y,z) = \frac{\frac{1}{\lambda_{xy}}+\frac{1}{\lambda_{yz}}-\frac{1}{\lambda_{xz}}}{\frac{2}{\sqrt{\lambda_{xy} \lambda{yz}}}}$$

that:

$$\pi = \arccos \left ( \frac{\frac{1}{\lambda_{xy}}+\frac{1}{\lambda_{yz}}-\frac{1}{\lambda_{xz}}}{\frac{2}{\sqrt{\lambda_{xy} \lambda_{yz}}}}\right ) +\arccos \left ( \frac{\frac{1}{\lambda_{zx}}+\frac{1}{\lambda_{xy}}-\frac{1}{\lambda_{zy}}}{\frac{2}{\sqrt{\lambda_{zx} \lambda_{xy}}}}\right ) +\arccos \left ( \frac{\frac{1}{\lambda_{yz}}+\frac{1}{\lambda_{zx}}-\frac{1}{\lambda_{yx}}}{\frac{2}{\sqrt{\lambda_{yz} \lambda_{zx}}}} \right ) $$

where $\lambda_{ab}$ is the wavelength defined by the Rydberg formula:

$$1/\lambda_{ab} = R_H |1/a^2-1/b^2|$$

and $x,y,z$ are natural numbers.

A further possible application to physics might be given as described in the answer to this question: Simplex invariants?

n= 2
S  = range(1, 2)
Vector =  (-1.00000000000000)
Vardet: 0 = $ -c_{12}^{2} + 1 $
det =  0.000000000000000
n= 3
S  = range(1, 3)
Vector =  (-0.800000000000000, -0.316227766016838, -0.316227766016838)
Vardet: 0 = $ 2 \, c_{12} c_{13} c_{23} - c_{12}^{2} - c_{13}^{2} - c_{23}^{2} + 1 $
det =  1.94289029309402e-16
n= 4
S  = range(1, 4)
Vector =  (-0.611130643138949, 0.248069469178417, -0.509802390301733, -0.860054653873699, 0.250892341323851, -0.375447149783241)
Vardet: 0 = $ c_{14}^{2} c_{23}^{2} - 2 \, c_{13} c_{14} c_{23} c_{24} + c_{13}^{2} c_{24}^{2} - 2 \, c_{12} c_{14} c_{23} c_{34} - 2 \, c_{12} c_{13} c_{24} c_{34} + c_{12}^{2} c_{34}^{2} + 2 \, c_{12} c_{13} c_{23} + 2 \, c_{12} c_{14} c_{24} + 2 \, c_{13} c_{14} c_{34} + 2 \, c_{23} c_{24} c_{34} - c_{12}^{2} - c_{13}^{2} - c_{14}^{2} - c_{23}^{2} - c_{24}^{2} - c_{34}^{2} + 1 $
det =  0.000000000000000

Here $n$ counts the number of distinct principal quantum number of an energy level and $\det=0$ gives as in the answer above an algebraic variety which should be satisfied by the $\lambda_{ij}$.

Example for $n=3$ we get the Cayley nodal cubic which is studied in algebraic geometry:

$$0 = 2 \, c_{12} c_{13} c_{23} - c_{12}^{2} - c_{13}^{2} - c_{23}^{2} + 1 $$

where $c_{12} = S(x,y,z), c_{13} = S(z,x,y), c_{23} = S(y,z,x)$

Here is a picture of it:

Example for $n=4$ the algebraic variety is given by:

$0=c_{14}^{2} c_{23}^{2} - 2 \, c_{13} c_{14} c_{23} c_{24} + c_{13}^{2} c_{24}^{2} - 2 \, c_{12} c_{14} c_{23} c_{34} - 2 \, c_{12} c_{13} c_{24} c_{34} + c_{12}^{2} c_{34}^{2} + 2 \, c_{12} c_{13} c_{23} + 2 \, c_{12} c_{14} c_{24} + 2 \, c_{13} c_{14} c_{34} + 2 \, c_{23} c_{24} c_{34} - c_{12}^{2} - c_{13}^{2} - c_{14}^{2} - c_{23}^{2} - c_{24}^{2} - c_{34}^{2} + 1 $

Let $\phi(a) = \frac{1}{a^2}\sum_{i=1}^a \sqrt{2i-1}e_i$ be an embedding of the natural number $a$ into the Hilbert space $l_2$ ( We might call this one feature vector of the natural number $a$).

Then it follows that:

$$\left < \phi(a),\phi(b) \right > = \frac{1}{a^2b^2} \sum_{i=1}^{\min(a,b)} (2i-1) = \frac{\min(a,b)^2}{a^2b^2}= \frac{\min(a,b)^2}{\min(a,b)^2\max(a,b)^2}= \frac{1}{\max(a,b)^2} = k(a,b)$$

The Lyman series and similarily the other following series might be visualized as:

In this visualization we use the fact that:

$$d(a,b)^2 + \frac{1}{a^2} = \frac{1}{b^2},\text{ if } b < a$$

which can be immediately recognized as Pythagoras theorem, since $|\phi(a)| = \frac{1}{a}$. One can also recognize the circle of Thales in this context of visualization. The angle $\rho$ between $n=a$ and $n=b$ used in this visualizations, is given by:

$$\cos(\rho) = \frac{k(a,b)}{\sqrt{k(a,a)k(b,b)}} = \frac{\frac{1}{\max(a,b)^2}}{\sqrt{\frac{1}{a^2b^2}}} = \frac{ab}{\max(a,b)^2} = \frac{\min(a,b)}{\max(a,b)}=\text{ one Jaccard kernel / similarity}$$

One can add, that doing kernel PCA with the given kernel $k(a,b)$ above, in two dimension, yields a similar, if not to say identical image as used in the visualizations above.

Edit with speculative hypothesis based on mathematical properties of the Rydberg formula:

Physics interpretation: The Rydberg formula:

$$1/\lambda_{ab} = R_H |1/a^2-1/b^2|, a \neq b , a,b \in \mathbb{N}$$

is symmetric in $a,b$. This suggests that it is possible to interchange $a$ for $b$ and vice versa in the hydrogen atom. This change is accompained by an absorption or emission of light of wavelength $\lambda_{ab}$.

Hypothesis:

H0) The hydrogen atom might be thought of as a Hilbert space $H$ where the natural numbers are embedded with $\phi$, so that $\left< \phi(a) , \phi(b) \right > = k(a,b) = \frac{1}{\max(a,b)^2}$.

H1) One can only physically measure the distance $d(a,b) = \sqrt{|1/a^2-1/b^2|}$ between two feature vectors $\phi(a),\phi(b)$ of natural numbers $a,b$ in the hydrogen atom.

H2) The hydrogen atom maintains a state $\phi(n_1),\phi(n_2),\cdots,$, which might be thought of as a permutation of the feature vectors of natural numbers $\phi(1),\phi(2),\phi(3),\cdots$.

H3) Every physical measurment of $d(a,b)$ swaps $\phi(a)$ and $\phi(b)$ in the state of the hydrogen atom.

H4) The feature vector $\phi(x)$ of a natural number $x$, cannot be measured.

Comment: All these hypothesis stem from the Hilbert-space interpretation of the Rydberg formula.

Since it is observed that by the law of cosine, we can associate to each tupel $(x,y,z)$ of natural numbers, an angle $[x,y,z]$ in a triangle spanned by the vectors $\phi(x),\phi(y),\phi(z)$, where $[x,y,z] := S(x,y,z) := \frac{\frac{1}{\lambda_{xy}}+\frac{1}{\lambda_{yz}}-\frac{1}{\lambda_{xz}}}{\frac{2}{\sqrt{\lambda_{xy} \lambda{yz}}}}$ is the angle at $\phi(y)$.

We give those three angles names:

$$\alpha = [z,x,y], \beta = [x,y,z], \gamma = [y,z,x]$$

Now measuring the distance $d(x,y)$ in an experiment induces a swapping of $\phi(x)$ and $\phi(y)$ with a transposition $\tau = (x,y)$, will give us for every other natural number $z$:

$$ \tau \cdot \alpha = 2 \pi - \beta$$ $$ \tau \cdot \beta = 2 \pi - \alpha$$ $$ \tau \cdot \gamma = 2 \pi - \gamma$$

Applying $\cos$ to the last equations we get:

$$\cos(\tau \cdot \alpha) = \cos(\beta),\cos( \tau \cdot \beta) = \cos(\alpha) , \cos( \tau \cdot \gamma) = \cos(\gamma) $$. Hence one might say that measuring $d(x,y)$ swaps not only the $\phi(x)$ and $\phi(y)$ but also the cosines of the corresponding angles in the Hilbert space.

For instance let at time $t$ be $\pi_t$ the state of the hydrogen atom. Then we measure $d(x,y),d(y,z),d(x,z)$ so we get after three measurements the new state $\pi_{\hat{t}}$:

$$\pi_{\hat{t}} = (x,z,y) \cdot \pi_t = (x,z)(y,z)(x,y) \cdot \pi_t$$

If on the other hand we would have measured $d(x,y),d(x,z),d(y,z)$ we would have gotten the new state $\pi_{\hat{t}}$:

$$\pi_{\hat{t}} = (x,z) \cdot \pi_t = (y,z)(x,z)(x,y) \cdot \pi_t$$

which has the same effect on $\pi_t$ as just measuring $d(x,z)$ in one step.

Possibly related question where it is thought about applications of p.d. kernel on natural numbers in physics and the space of Gram matrices is a Riemann manifold.

Answer 1

Since the question asks about a "physical interpretation", let me record the full expression for the wavelength $\lambda$ of the hydrogen spectral lines, which depends on four (not two) integers $n_1,n_2,j_1,j_2$,

$$\frac{1}{\lambda} = R_\text{H}\left(E_{n_1,j_1}-E_{n_2,j_2}\right)$$ $$E_{n,j} = -\frac{2}{\alpha^2}\left[ 1 - \left( 1 + \left[ \frac{\alpha}{n - j - \frac{1}{2} + \sqrt{\left( j + \frac{1}{2} \right)^2 - \alpha^2}} \right]^2 \right)^{-1 / 2}\right]. $$ It so happens that $\alpha=0.00729735\cdots$ is small, and hence $E_{n,j}\approx 1/n^2$, but I don't think that any "inner product" interpretation can shed light on this formula.

A somewhat more promising line of approach to arrive at the spectral properties of atoms using purely mathematical reasoning (without physics input) is described in Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?