Abstract
Formulas of quantum mechanics often contain Planck’s constant h divided by a mass m. Putting in the choice of mass used throughout this corpus, m = 4π, and using h = 2πħ, gives h/m = ħ/2. This paper counts which formulas lose their π, what the remaining 2 is, and whether this ħ/2 is the same thing as the ħ/2 of the uncertainty bound. No new theorem or law is claimed.
First, h/m is a single quantity. For the electron, h divided by the mass, the Compton wavelength times the speed of light, and twice the “quantum of circulation” of the table of constants are all $\mathbf{7.2738950934}$×10⁻⁴ m²/s; the relative differences among the table values were at most 6.566×10⁻¹². The same quantity sits in the formulas for wavelengths and in the formula for vortices.
Second, π drops out of the formulas that count in full turns. With m = 4π and h = 2πħ, the de Broglie wavelength becomes ħ/(2v), the Compton wavelength ħ/(2c), the quantum of circulation ħ/2, and the thermal wavelength ħ divided by the square root of 2kT. None of the eight formulas examined keeps a π. On the other hand, formulas in which ħ faces the mass (five of them, such as the spreading coefficient ħ/(2m)) keep a 4π.
Third, “setting the mass to 4π” means taking one part in 4π of the particle’s mass as the unit of mass. For a gas of rubidium 87 at a number density of 10²⁰ m⁻³, the condensation temperature computed from the usual formula and from the formula without π in the re-chosen unit was $\mathbf{398.3307}$ nK in both cases. Only the appearance of the formula changes; no measurable value does.
Fourth, the remaining 2 is the surface area of the sphere divided by the circumference of the circle. With the mass taken as the surface area of the unit sphere in d dimensions, h/m is ħ divided by “the volume of the unit ball two dimensions lower”: 1, 1/2, 1/π, 3/(4π) times ħ from two dimensions upward. If points are placed uniformly on a sphere and two coordinates are dropped, the rest falls uniformly inside the ball two dimensions lower. In three dimensions this is Archimedes’ theorem on the zones of a sphere; checked with random points, it was uniform within 1.67 standard errors in each of three to six dimensions (the control, dropping only one coordinate, was off by more than 300 standard errors).
Fifth, and this is the core of the paper: this ħ/2 is the same number as the ħ/2 of the uncertainty bound, but it has a different root. The uncertainty bound is a statement about one axis; measured on a Gaussian wave packet it is $\mathbf{0.500000000000}$ whatever the width, and it does not move when the dimension of space is changed. The value from h/m moves: $\mathbf{1.000000}$, $\mathbf{0.500000}$, $\mathbf{0.318310}$, $\mathbf{0.238732}$, $\mathbf{0.202642}$ in two to six dimensions. The two are the same number only in three dimensions.
Sixth, numerical checks. The coefficient ħ/(2m) obtained from the spreading of a wave packet with ħ = 1 and m = 4π is $\mathbf{0.039788736}$, that is 1/(8π), and 4π times it is $\mathbf{0.500000000}$. The circulation around a vortex was $0.500000000$, $1.000000000$ and $1.500000000$ for winding numbers 1, 2 and 3. In the choice of units of this corpus with ħ = 1/(4π), h itself is $\mathbf{0.500000000}$.
To be candid, m = 4π is a choice of unit and adds no physical meaning to the form ħ/2. The 2π inside h is one turn of the phase, and the 2π inside the surface area of the sphere is the circumference of a circle. Paper 18 treated the two as different roots. h/m = ħ/2 is the form in which the two cancel as numbers.
Keywords: Planck constant, de Broglie wavelength, Compton wavelength, quantum of circulation, choice of unit, dimension, uncertainty.
Scope (what is not claimed)
No new theorem or law is claimed. The de Broglie wavelength (de Broglie 1925), the Compton wavelength (Compton 1923), the fact that the circulation of a vortex is an integer multiple of h/m (Onsager 1949), the thermal wavelength, and the fact that the area of a zone of a sphere depends only on its height (Archimedes) are all known. Setting the mass to 4π is not called a fact about nature. It is a choice of the unit of mass. No measurable value changes. No meaning is added to the form ħ/2. That it is the same number as the uncertainty bound is not written as a common root. The 1/2 of spin is not treated. No claim is made that quantum mechanics exists in higher dimensions. The formula is only listed against the dimension. Measurement is not discussed. The values of the table of constants (CODATA 2022) are used only as input, and the agreement among table values is a consequence of how the table is made. Relation to earlier papers: Paper 18 showed that putting m = 4π into the kinetic energy gives 2πv², and counted that the same number 2π comes from three different roots: the solid angle of a hemisphere, the period of a phase, and the circumference of a circle. Here the same substitution is made in h/m, and it is seen that the 2π of h (the period of the phase) is cancelled and a 2 remains. Paper 129 showed that the 4π in the uncertainty bound ħ/2 = h/(4π) is the product of 2 and 2π and not a solid angle, and that the bound does not move when the dimension of space is changed. Here it is counted that the ħ/2 appearing in h/m moves with the dimension, so that the two have different roots. Paper 912 showed that taking the mass as the surface area of the unit sphere in d dimensions makes the coefficient of temperature the volume of the unit ball in every dimension. Here the same choice makes the volume of the unit ball two dimensions lower appear in h/m. Paper 115 showed that the recursion for the volume of the sphere goes in steps of two. The relation used here, surface area = 2π × volume two dimensions lower, is one such step. Paper 59 showed that the volume of the unit ball is greatest in five dimensions. As its reverse side, the family of h/m is smallest in seven dimensions. Paper 663 counted the condensation temperature and the thermal wavelength of the ideal Bose gas. Here those values are reproduced from the formula in the re-chosen unit. Paper 39 treated the choice ħ = 1/(4π). In that choice h is 1/2. Paper 60 showed that the 4π in E = 4πc² is a mass placed there by a choice of unit. Here too the 4π is a placed mass. What is added: listing what happens to h/m under m = 4π, separated into formulas that lose π and formulas that keep it, showing by numbers that the substitution is a re-choice of the unit of mass, listing the remaining 2 against the dimension as the ratio of the surface area of the sphere to the circumference of the circle, and separating it from the ħ/2 of the uncertainty bound by the dimension.
1. The question
Planck’s constant is written in two ways. $h$ counts per full turn, and $\hbar=h/2\pi$ counts per radian. In the formulas of quantum mechanics $h$ often appears divided by a mass $m$. Examples are the wavelength of a moving particle $h/(mv)$, the wavelength proper to a particle at rest $h/(mc)$, and the smallest unit of the strength of a vortex $h/m$.
This corpus has used the choice of mass $m=4\pi$. Putting it in gives
$$\frac{h}{m} = \frac{2\pi\hbar}{4\pi} = \frac{\hbar}{2} .$$As a substitution it takes one line. Three things are to be counted. Which formulas lose their $\pi$, and which keep it? What number is the remaining 2? And since $\hbar/2$ is also the uncertainty bound, are the two the same thing?
2. $h/m$ is a single quantity
First, that $h/m$ is one quantity called by several names is checked with the values of the table of constants (CODATA 2022). The electron is taken as the example.
| Quantity | Value (m$^2$/s) |
|---|---|
| $h$ divided by the electron mass | $7.2738950934\times10^{-4}$ |
| Compton wavelength $\times$ speed of light | $7.2738950934\times10^{-4}$ |
| Twice the “quantum of circulation” of the table | $7.2738950934\times10^{-4}$ |
The Compton wavelength of the table is $2.42631023538\times10^{-12}$ m. The relative difference from $h$ divided by the mass was $-1.306\times10^{-13}$ for the Compton wavelength times the speed of light and $6.566\times10^{-12}$ for twice the quantum of circulation. The table lists the “quantum of circulation” as $h/(2m)$. A wavelength times a speed, and the strength of the flow once around a vortex, are one and the same quantity.
3. Setting the mass to $4\pi$: formulas that lose $\pi$ and formulas that keep it
$m=4\pi$ and $h=2\pi\hbar$ were put into common formulas and simplified (symbolic computation). First, the formulas in which $h$ faces the mass.
| Quantity | Usual form | Form with $m=4\pi$ |
|---|---|---|
| de Broglie wavelength | $h/(mv)$ | $\hbar/(2v)$ |
| Compton wavelength | $h/(mc)$ | $\hbar/(2c)$ |
| Quantum of circulation | $h/m$ | $\hbar/2$ |
| Thermal wavelength | $h/\sqrt{2\pi mk_{\rm B}T}$ | $\hbar/\sqrt{2k_{\rm B}T}$ |
| Coefficient in the condensation temperature | $2\pi\hbar^2/m$ | $\hbar^2/2$ |
| Quantum concentration | $(mk_{\rm B}T/2\pi\hbar^2)^{3/2}$ | $(2k_{\rm B}T/\hbar^2)^{3/2}$ |
| Coefficient of the free propagator | $m/(2\pi\hbar t)$ | $2/(\hbar t)$ |
| Coefficient of the area of spreading | $4\pi\times\hbar/(2m)$ | $\hbar/2$ |
In all eight, not a single $\pi$ remains. In the thermal wavelength three $\pi$ go at once: the $2\pi$ of $h$, the $2\pi$ of the Gaussian integral and the $4\pi$ of the mass. Next, the formulas in which $\hbar$ faces the mass.
| Quantity | Usual form | Form with $m=4\pi$ |
|---|---|---|
| Spreading coefficient of a wave | $\hbar/(2m)$ | $\hbar/(8\pi)$ |
| Square of the oscillator length | $\hbar/(m\omega)$ | $\hbar/(4\pi\omega)$ |
| Kinetic energy | $\hbar^2k^2/(2m)$ | $\hbar^2k^2/(8\pi)$ |
| Bohr radius (Gaussian units) | $\hbar^2/(me^2)$ | $\hbar^2/(4\pi e^2)$ |
| Coefficient in the bound on the velocity uncertainty | $\hbar/(2m)$ | $\hbar/(8\pi)$ |
These five keep the $4\pi$ as it is. The difference is whether the formula counts angles in full turns or in radians. A formula that counts in full turns carries one $2\pi$, and that cancels against half of the $4\pi$ of the mass. A formula that counts in radians has nothing to cancel against.
4. What “setting to $4\pi$” amounts to
What does one do in setting the mass to $4\pi$? One takes one part in $4\pi$ of the particle’s mass as the unit of mass. Since only the unit has been re-chosen, no measurable value should change. This is checked with the condensation temperature of the ideal Bose gas counted in Paper 663. Take rubidium 87 ($86.909180527$ u) at a number density of $10^{20}$ m$^{-3}$.
| Way of computing | Condensation temperature |
|---|---|
| Usual formula $(2\pi\hbar^2/m)(n/\zeta)^{2/3}/k_{\rm B}$ | $398.3307$ nK |
| Formula without $\pi$, $(\hbar^2/2)(n/\zeta)^{2/3}/k_{\rm B}$, with the unit of mass $m/4\pi$ | $398.3307$ nK |
$\zeta$ is the value of the Riemann zeta function at $3/2$, $2.6123753487$, and the unit of mass comes to $1.148431\times10^{-26}$ kg. The ratio of the two values was $1.000000000000$. The thermal wavelength at the condensation temperature is $2.967189\times10^{-7}$ m from either formula, and the number density times the cube of the wavelength is $2.6123753487$ (a reproduction of the values of Paper 663). The formula without $\pi$ is not a new formula. It is the same formula written in another unit.
5. The remaining 2 is the surface area of the sphere divided by the circumference of the circle
The 2 in $h/m=\hbar/2$ is $4\pi$ divided by $2\pi$. $4\pi$ is the surface area of the unit sphere and $2\pi$ the circumference of the unit circle, so the 2 is “surface area of the sphere $\div$ circumference”. Whether this number is particular to three dimensions is examined by changing the dimension. Following Paper 18 and Paper 912, the mass is taken as the surface area $S_{d-1}$ of the unit sphere in $d$ dimensions.
| $d$ | $S_{d-1}$ | $V_{d-2}$ | $h/m\div\hbar=2\pi/S_{d-1}$ |
|---|---|---|---|
| 2 | $6.283185307$ | $1.000000000$ | $1.000000000$ |
| 3 | $12.566370614$ | $2.000000000$ | $0.500000000$ |
| 4 | $19.739208802$ | $3.141592654$ | $0.318309886$ |
| 5 | $26.318945070$ | $4.188790205$ | $0.238732415$ |
| 6 | $31.006276680$ | $4.934802201$ | $0.202642367$ |
| 7 | $33.073361792$ | $5.263789014$ | $0.189977219$ |
| 8 | $32.469697011$ | $5.167712780$ | $0.193509207$ |
$V_{d-2}$ is the volume of the unit ball two dimensions lower. In every dimension $S_{d-1}=2\pi V_{d-2}$ holds, so
$$\frac{h}{m} = \frac{2\pi\hbar}{S_{d-1}} = \frac{\hbar}{V_{d-2}} .$$In terms of $\pi$ the values are 1, $1/2$, $1/\pi$, $3/(4\pi)$ and $2/\pi^2$ times $\hbar$ from two dimensions upward. The 2 of three dimensions is the “volume” of the one-dimensional ball (a segment of length 2), that is, the diameter. Paper 59 showed that the volume of the unit ball is greatest in five dimensions. As its reverse side, this family is smallest, $0.189977219$, in seven dimensions.
There is a picture behind $S_{d-1}=2\pi V_{d-2}$. Place points uniformly on a sphere and forget two of the coordinates. The remaining coordinates are then scattered uniformly inside the ball two dimensions lower. The two forgotten coordinates run around a circle, so the size of the sphere is “the $2\pi$ of a circumference” times “the volume of the ball two dimensions lower”. In three dimensions this is Archimedes’ theorem that the area of a zone of a sphere depends only on the height of the zone (Heath 1897). It was checked with random points (20 batches of 200,000 points; the fraction falling within $0.5$ of the centre).
| $d$ | Dropping two | If uniform | Deviation (standard errors) | Dropping one (control) | If uniform |
|---|---|---|---|---|---|
| 3 | $0.500336$ | $0.500000$ | $+1.67$ | $0.133894$ | $0.250000$ |
| 4 | $0.249821$ | $0.250000$ | $-0.75$ | $0.057559$ | $0.125000$ |
| 5 | $0.124776$ | $0.125000$ | $-1.38$ | $0.025771$ | $0.062500$ |
| 6 | $0.062574$ | $0.062500$ | $+0.53$ | $0.011740$ | $0.031250$ |
When two coordinates are dropped, the value agreed with the uniform one within $1.67$ standard errors in every dimension. The control, dropping only one, was off by $305.9$ to $585.1$ standard errors. The distribution is uniform when exactly two coordinates are dropped. Paper 115 showed that the recursion for the volume of the sphere goes in steps of two. The relation used here is one such step.
6. It has a different root from the $\hbar/2$ of the uncertainty bound
$\hbar/2$ is also the lower bound on the product of the uncertainties of position and momentum. Is the $\hbar/2$ that appeared in $h/m$ the same thing? Paper 129 showed that the $4\pi$ in the uncertainty bound $\hbar/2=h/(4\pi)$ is the product of 2 and $2\pi$ and not a solid angle, and that the bound does not move when the dimension of space is changed. This is because the bound is a statement about one axis. Placing a Gaussian wave packet on a grid and measuring, the product of the width in position and the width in wavenumber was $0.500000000000$ for each of the widths $0.5$, $1.0$ and $3.0$.
| $d$ | $h/m\div\hbar$ (mass taken as $S_{d-1}$) | Uncertainty bound $\div\hbar$ |
|---|---|---|
| 2 | $1.000000$ | $1/2$ |
| 3 | $0.500000$ | $1/2$ |
| 4 | $0.318310$ | $1/2$ |
| 5 | $0.238732$ | $1/2$ |
| 6 | $0.202642$ | $1/2$ |
When the dimension is changed, only the left column moves. The two are the same number only in three dimensions. Since one can be moved without moving the other, the two come from different roots. The 2 of the uncertainty bound is fixed by the shape of the wave packet for which equality holds (the Gaussian). The 2 of $h/m$ is the ratio of the surface area of the sphere to the circumference of the circle.
7. The spreading of a wave and the circulation of a vortex
One formula from each of the two tables was checked numerically ($\hbar=1$). First, $\hbar/(2m)$ on the side that counts in radians. For a free particle, the squared width of a Gaussian wave packet grows in a time $t$ by $(\hbar t/(2m\sigma_0))^2$, with $\sigma_0$ the initial width. Advancing the packet on a grid, measuring the width and solving back for the coefficient $\hbar/(2m)$ gave $0.500000000$ for $m=1$ and $0.039788736$ for $m=4\pi$. The latter is $1/(8\pi)$: the $4\pi$ is still there. Multiplying it by $4\pi$ gives $0.500000000$, back to $h/m=\hbar/2$.
Next, $h/m$ on the side that counts in full turns. The speed of the flow around a vortex is the gradient of the phase of the wave times $\hbar/m$, and the speed summed along a closed path (the circulation) is fixed by how many times the phase turns while going once around the path. The advance of the phase was summed along a square path.
| Winding number | Circulation ($m=1$) | Circulation ($m=4\pi$) | Path not enclosing the vortex ($m=4\pi$) |
|---|---|---|---|
| 1 | $6.283185307$ | $0.500000000$ | $-5.654\times10^{-16}$ |
| 2 | $12.566370614$ | $1.000000000$ | $1.626\times10^{-15}$ |
| 3 | $18.849555922$ | $1.500000000$ | $-1.626\times10^{-15}$ |
For $m=1$ the circulation is an integer multiple of $2\pi$, and for $m=4\pi$ an integer multiple of $1/2$. The $2\pi$ appearing here is the angle of one turn of the phase. What it means for $h$ to be $2\pi$ times $\hbar$ shows up directly in the numbers.
This corpus also has a choice in which $\hbar=1/(4\pi)$ (Paper 39). In that choice $\hbar=0.079577472$ and $h=2\pi\hbar$ is $0.500000000$. Planck’s constant itself is $1/2$. Together with $m=4\pi$, $h/m=0.039788736$, which is $1/(8\pi)$. This is the same number as the coefficient of the wave packet above, but it is a different quantity in a different choice of units.
8. Relation to earlier papers
Paper 18 showed that putting $m=4\pi$ into the kinetic energy gives $2\pi v^2$, and counted that the same number $2\pi$ comes from three different roots: the solid angle of a hemisphere, the period of a phase, and the circumference of a circle. The $2\pi$ of $h=2\pi\hbar$ belongs to the period of a phase. In the formula of this paper, $h/m=\hbar/V_{d-2}$, this $2\pi$ cancels against the $2\pi$ inside the surface area of the sphere, $S_{d-1}=2\pi V_{d-2}$. The latter is the circumference of the circle around which the two forgotten coordinates run. Paper 18 treated the period of a phase and the circumference of a circle as different roots. The present paper follows that distinction and writes the cancellation as one “of numbers”. Whether the two can be read as the same root is not decided here.
Paper 912 showed that taking the mass as $S_{d-1}$ makes the coefficient of temperature the volume $V_d$ of the unit ball in every dimension. With the same choice, $h/m$ brings out $V_{d-2}$, two dimensions lower. The pattern, a mass taken as a surface area turning a coefficient into a volume, is the same; the dimension of the volume that appears differs.
Paper 60 showed that the $4\pi$ in $E=4\pi c^2$ is not a solid angle but a mass placed there by a choice of unit. The $4\pi$ of this paper has the same standing. $h/m=\hbar/2$ does not mean that a quantity $\hbar/2$ has been newly found in nature. It means that with the unit of mass so chosen, $\pi$ becomes invisible in the formulas that count in full turns.
Closing
Closing: setting the mass to 4π turns h/m into ħ/2. h/m is the same single quantity as the Compton wavelength times the speed of light and as twice the quantum of circulation; for the electron it is $\mathbf{7.2738950934}$×10⁻⁴ m²/s. π drops out of the eight formulas that count in full turns, and a 4π remains in the five that count in radians. The substitution is a re-choice of the unit of mass, and the condensation temperature was $\mathbf{398.3307}$ nK from either formula. The remaining 2 is the surface area of the sphere divided by the circumference of the circle; with the mass taken as the surface area of the sphere in d dimensions, the form is ħ divided by the volume of the ball two dimensions lower. Its value moves with the dimension, $\mathbf{1.000000}$, $\mathbf{0.500000}$, $\mathbf{0.318310}$, while the 1/2 of the uncertainty bound does not. To be candid, the shared form ħ/2 does not connect h/m with the uncertainty bound. They are the same number only in three dimensions; one comes from the ratio of a sphere to a circle, the other from the shape of a wave packet. The separator is whether it moves when the dimension is changed.
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Reference
Yamagishi, Y. (2026). Setting m = 4π Turns h/m into ħ/2, and Its 2 Has a Different Root from the 2 of the Uncertainty Bound [Paper 1083]. YOI: https://www.yamagishiyuuki.com/yoi/1083
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1083. This paper is a preprint and has not been peer reviewed.