Abstract
Paper 01 showed that the weight of one spherical skin (the area 4πr² times the density) is the same at every radius only when the density is 1/r². What do we see if we apply the same ruler to the electron cloud inside an atom? The electron cloud is not 1/r². This paper counts how it departs. No new theorem or law is claimed.
First, the peaks of the weight per skin sit where the local exponent of the cloud is exactly 2. Call the local exponent the n obtained by viewing the density near a radius as 1/rⁿ. The weight per skin has a peak exactly when the local exponent is 2, and this is an identity. Searching all peaks of 1s, 2s, 2p, 3s, 3p, 3d, 4s and 4f, the local exponent at every peak was 2.000000 (Section 2).
Second, the 1s peak is the Bohr radius. The local exponent of 1s is 2r (with the Bohr radius as 1), and it equals 2 only at r = 1. The probability of being inside the Bohr radius is 0.3233236, and the mean radius is 1.5 (Section 3).
Third, and this is the core of the paper: the peak of a circular state is Bohr’s radius n². In the states of largest angular momentum (l = n - 1), the radius where the local exponent equals 2 coincides, for every n, with n², the value Bohr set as the orbit radius in 1913. The spread divided by the mean shrinks from 0.577350 at n = 1 to 0.070535 at n = 100, and the probability of lying inside the peak approaches 1/2 (0.481206 at n = 100) (Section 4).
Fourth, the two peaks of 2s can be written with the golden ratio. The peaks are 2/φ² = 0.763932 and 2φ² = 5.236068; their product is 4 and their geometric mean is the node at r = 2. On a logarithmic scale the two peaks lie the same 0.962423650 away from the node (Section 5).
Fifth, the separator is whether the exponent 2 holds globally or locally. A cloud whose density is 1/r² everywhere puts the same weight on every skin and has neither a peak nor a size. The cloud of an atom crosses a local exponent of 2 exactly once, and that radius becomes the size of the atom. The electron does not move with a 1/r² density, but the 2 of the uniqueness theorem remains inside the cloud as the one point that sets the size of the atom (Section 6).
Keywords: hydrogen atom, radial probability, Bohr radius, uniqueness theorem, local exponent, golden ratio, separator.
Scope (what is not claimed)
No new theorem or law is claimed. The radial functions of the hydrogen atom, the most probable radius, the fact that the peak of a circular state is at n², and the approach to classical orbits at large n are all standard. What this paper adds is applying the ruler of Paper 01 to the electron cloud and restating the position of the peaks through the local exponent. It does not enter into the interpretation of where the electron really is, and it does not say that Bohr’s model is correct. The nucleus is held fixed; screening in many-electron atoms, relativistic corrections and the size of the nucleus are not included. The local exponent is just the derivative of the logarithm of the density with respect to the logarithm of the radius, and is not called a new physical quantity.
1. The question
Paper 01 showed that, for a spherically symmetric density $\rho(r)=r^{-n}$, the weight of one skin of thickness $dr$, $4\pi r^{2}\rho dr$, is the same at every radius only when $n=2$. Paper 1092 counted that extending this 2 to other dimensions gives $d-1$.
The electron cloud is not $1/r^{2}$. The density of the lowest state of the hydrogen atom (1s) is $|\psi|^{2}\propto e^{-2r}$: flat near the nucleus and falling exponentially far away (below, the Bohr radius is the unit of length). Question: applying the same ruler of skins, what do we see in the electron cloud?
2. The ruler and an identity
With the radial function $R_{nl}(r)$, the weight (probability) per skin summed over angles is $P(r)=r^{2}R_{nl}(r)^{2}$, with $\int_{0}^{\infty}P dr=1$. Viewing the density $\rho=|\psi|^{2}$ near a radius as $r^{-n}$, call that $n$ the local exponent:
$$n_{\mathrm{eff}}(r)=-\frac{d\ln\rho}{d\ln r}.$$Then
$$\frac{d}{dr}\bigl(r^{2}\rho\bigr)=r\rho\,\bigl(2-n_{\mathrm{eff}}(r)\bigr).$$So where $\rho>0$, the weight per skin is stationary only when $n_{\mathrm{eff}}=2$. This is an identity and is not limited to hydrogen. The $n=2$ of Paper 01 (flat skins everywhere) is the case where $n_{\mathrm{eff}}$ is 2 at every radius.
Observation 2.1 For eight states of hydrogen, all stationary points of the weight per skin were searched. Every stationary point with $\rho>0$ is a peak, and the local exponent there is 2.000000 in every case. The remaining stationary points are nodes ($\rho=0$), where the local exponent is undefined.
| State | Peak radius (local exponent) | Node radius |
|---|---|---|
| 1s | 1.000000 (2.000000) | none |
| 2s | 0.763932, 5.236068 (both 2.000000) | 2.000000 |
| 2p | 4.000000 (2.000000) | none |
| 3s | 0.740037, 4.185930, 13.074033 (all 2.000000) | 1.901924, 7.098076 |
| 3p | 3.000000, 12.000000 (both 2.000000) | 6.000000 |
| 3d | 9.000000 (2.000000) | none |
| 4s | 0.732344, 4.000000, 10.649564, 24.618092 (all 2.000000) | 1.871644, 6.610815, 15.517541 |
| 4f | 16.000000 (2.000000) | none |
3. 1s: the peak is the Bohr radius
For 1s, $\rho\propto e^{-2r}$, so $n_{\mathrm{eff}}(r)=2r$. It is 1 at radius 0.5, 2 at radius 1 and 4 at radius 2, growing in proportion to the radius. It equals 2 only at $r=1$, which is the peak of the weight per skin $P(r)=4r^{2}e^{-2r}$, that is, the Bohr radius.
Observation 3.1 The probability of being inside the Bohr radius is $1-5e^{-2}=$ 0.3233236. The mean radius is $3/2$, shifted outward from the peak. Near the nucleus the local exponent of the cloud is close to 0 (flat density), and the weight per skin grows as $r^{2}$, heavier outward. Far away the exponent is large, heavier inward. In between, the one point where the local exponent passes exactly through the 2 of the uniqueness theorem becomes the size of the atom.
4. Circular states: the peak is Bohr’s radius $n^{2}$
In the states of largest angular momentum ($l=n-1$), $P(r)\propto r^{2n}e^{-2r/n}$. The local exponent is
$$n_{\mathrm{eff}}(r)=\frac{2r}{n}-2(n-1),$$which for $n=1,2,3,4$ is $2r$, $r-2$, $2r/3-4$ and $r/2-6$. It equals 2 at $r=n^{2}$, which coincides for every $n$ with the value Bohr set as the orbit radius in 1913 ($n^{2}$ times the Bohr radius).
Since $P$ is a gamma distribution with shape $2n+1$ and scale $n/2$, the mean is $n^{2}+n/2$ and the spread (standard deviation) divided by the mean is $1/\sqrt{2n+1}$.
| $n$ | Peak radius | Mean radius | Spread / mean | Probability inside the peak |
|---|---|---|---|---|
| 1 | 1 | 1.5 | 0.577350 | 0.323324 |
| 2 | 4 | 5 | 0.447214 | 0.371163 |
| 3 | 9 | 10.5 | 0.377964 | 0.393697 |
| 4 | 16 | 18 | 0.333333 | 0.407453 |
| 5 | 25 | 27.5 | 0.301511 | 0.416960 |
| 6 | 36 | 39 | 0.277350 | 0.424035 |
| 10 | 100 | 105 | 0.218218 | 0.440907 |
| 100 | 10000 | 10050 | 0.070535 | 0.481206 |
Observation 4.1 As $n$ grows, the width of the cloud shrinks relative to its mean ($n=100$: 0.070535), and the probability of lying inside the peak approaches $1/2$ ($n=100$: 0.481206). The cloud gathers onto a thin skin of radius $n^{2}$. Bohr’s orbit remains inside the cloud as the limiting form of this skin.
Observation 4.2 Paper 180 counted that the ratio of the frequency between adjacent levels of hydrogen to the classical orbital frequency approaches 1 as $1+3/(2n)$, showing an approach to the classical that means "large quantum number". The $1/\sqrt{2n+1}$ of this paper counts the same "large $n$" through spatial spread rather than frequency: a different quantity.
5. 2s: two peaks and the golden ratio
For 2s, $P\propto r^{2}(2-r)^{2}e^{-r}$, with a node at $r=2$. The peaks are the two roots $3\mp\sqrt5$ of $r^{2}-6r+4=0$. With the golden ratio $\varphi=(1+\sqrt5)/2$, $\varphi^{2}=(3+\sqrt5)/2$, so the two peaks are $2/\varphi^{2}=0.763932$ and $2\varphi^{2}=5.236068$.
Observation 5.1 The product of the two peaks is 4 and their geometric mean is 2, equal to the radius of the node. On a logarithmic scale the two peaks lie the same 0.962423650 away from the node. The golden ratio appears because the condition for a peak is quadratic and its coefficients produce $\sqrt5$. The system has treated the golden ratio elsewhere (14 papers mention it), but the $\varphi$ here comes from the quadratic for the 2s peaks; equal numbers are not called the same root (the criterion of Paper 300).
6. The separator: is the exponent 2 global or local?
| A cloud that is $1/r^{2}$ everywhere (Paper 01) | The electron cloud of hydrogen (this paper) | |
|---|---|---|
| Local exponent | 2 at every radius | $-2l$ near the nucleus (0 for s), growing far away |
| Weight per skin | the same for every skin (flat) | a peak at the radius where it crosses 2 |
| Size | undetermined | the peak radius (Bohr radius for 1s, $n^{2}$ for circular states) |
Observation 6.1 If the density is $1/r^{2}$ everywhere, every skin carries the same weight, so the cloud has no radius one can point to as "here". The cloud of an atom crosses a local exponent of 2 exactly once, and that radius becomes the size of the atom. The separator is whether the exponent 2 holds globally (everywhere) or locally (crossed at a single point).
Observation 6.2 Inside the atom, what is $1/r^{2}$ everywhere is not the electron cloud but the electric field of the nucleus (the same electric flux passes through every skin). The 2 of the uniqueness theorem lives in two places inside the atom: globally in the field, and as a single point in the electron cloud.
7. What the checks showed
The tool that searches for stationary points at first missed the peak of 1s. A dividing point fell exactly on $r=1$, where the derivative is 0, so it was not picked up as an interval where the sign changes. After fixing the tool to pick up a zero at a dividing point, the peak of 1s appeared at $r=1$ (local exponent 2.000000). Because the tool had first been tried on known values (the 1s peak at 1, the mean radius $3/2$, the 2p peak at 4), the miss was noticed.
Closing
Closing: the electron cloud is not the $1/r^{2}$ of the uniqueness theorem. Yet the weight per skin of the cloud peaks at the radius where the local exponent of the cloud is exactly 2, and this is an identity. The 1s peak is the Bohr radius, the peak of a circular state coincides with Bohr’s radius $n^{2}$ for every $n$, and the larger $n$ is, the more the cloud gathers onto that skin (spread divided by mean is 0.070535 at $n=100$). The two peaks of 2s are $2/\varphi^{2}$ and $2\varphi^{2}$, bracketing the node at $r=2$ exactly on a logarithmic scale. The separator is whether the exponent 2 is global or local. A cloud that is 2 everywhere has no size; a cloud that crosses 2 once has a size. The size of an atom can be written as the one point where the cloud passes through the 2 of the uniqueness theorem.
Revision Record
2nd Edition (October 2026): In the Japanese edition, Observation 5.1 said the golden ratio appears in the titles of 14 papers; this was corrected to "in the text of 14 papers", which is what the scan counted. The English text already said "14 papers mention it" and is unchanged apart from this record. No number or claim changes.
References
- Bohr, N. (1913). On the Constitution of Atoms and Molecules. Philosophical Magazine.
- Yamagishi, Y. (2026). The Uniqueness Theorem of n = 2 for Spherically Symmetric Density Distributions [Paper 01]. YOI: https://www.yamagishiyuuki.com/yoi/1; DOI: 10.5281/zenodo.21403186
- Yamagishi, Y. (2026). “The Classical Limit” Is Not One Limit [Paper 180]. YOI: https://www.yamagishiyuuki.com/yoi/180; DOI: 10.5281/zenodo.22099046
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The “2” of the Inverse-Square Density Is Two 2s, and the 2 of the Shell Mass Meets the 2 of the Horizon Balance Only in Three Dimensions [Paper 1092]. YOI: https://www.yamagishiyuuki.com/yoi/1092
Reference
Yamagishi, Y. (2026). The Weight per Skin of the Electron Cloud Peaks Where Its Local Exponent Is Exactly 2, and for Circular States That Radius Is Bohr’s n² [Paper 1096]. YOI: https://www.yamagishiyuuki.com/yoi/1096
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1096. This paper is a preprint and has not been peer reviewed.