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Equations involving Planck's constant, $h ,$ are often simplified by instead writing them in terms of the reduced Planck's constant, $\hbar \equiv \frac{h}{2 \pi}.$ But where did the symbol for the reduced Planck's constant, $\hbar ,$ come from?

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    $\begingroup$ Related Why was the reduced Planck constant introduced and when? $\endgroup$ – Conifold Sep 14 at 6:25
  • $\begingroup$ Personally I go for the theory that it was originated at a cattle ranch as their brand :-) $\endgroup$ – Carl Witthoft Sep 16 at 12:04
  • $\begingroup$ Note: I've tentatively accepted my own answer, but I'd be more than happy to accept a new answer that further clarifies. Please feel free to use the content from my answer as a starting place; further progress would probably involve checking out Dirac's personal notebooks/correspondence or a later retrospective article. $\endgroup$ – Nat Oct 3 at 18:13
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$ {\def\Target#1{\rlap{\smash{\label{#1}\phantom{\tag{#1}}}}}} {\def\BackUp{\raise{0.25em}{\Tiny{\boxed{\boldsymbol{\Uparrow} \hspace{-2px}}}}}} $tl;dr- It's unclear. The symbol $`` \hbar "$ itself wasn't anything new. Paul Dirac used it defining $\hbar \equiv \frac{h}{2 \pi}$ in a 1926 paper, but didn't explain the choice of the symbol. It might still be possible for someone to figure out the reason for this unusual symbol if they were to examine Dirac's personal notebooks or correspondence, or perhaps a later retrospective publication, but no explanation is apparently found in the original public appearances of $\hbar \equiv \frac{h}{2 \pi} .$


$\textbf{Timeline} \Target{Timeline}$

In short, while it seems reasonable to assume Dirac selected $`` \hbar "$ in part due to its similarity to $`` h " ,$ it's still unclear what else may've played into his choice. More information might be gleaned from Dirac's personal journals or correspondence.


$\BackUp$ $\textbf{Early history:}~~`` \mathbf{\hbar} " ~\textbf{appears in various old alphabets.} \Target{Early}$

The symbol itself, $\hbar ,$ is nothing new. Glancing at Wikipedia real quick, looks like it's earlier referenced as:

  1. in the Latin alphabet;

  2. the Slavic Cyrillic letter, Tshe;

  3. the astronomical symbol of Saturn;

  4. the alchemical symbol for lead.

Ħ (minuscule: ħ) is a letter of the Latin alphabet, derived from H with the addition of a bar. It is used in Maltese and in Tunisian Arabic transliteration (based on Maltese with additional letters) for a voiceless pharyngeal fricative consonant (corresponding to the letter heth of Semitic abjads). Lowercase ħ is used in the International Phonetic Alphabet for the same sound.

In quantum mechanics, an italic (U+210F) with a line, represents the reduced Planck constant. In this context, it is pronounced "h-bar".

The lowercase resembles the Cyrillic letter Tshe (ћ), or the astronomical symbol of Saturn (♄).

"H with stroke", Wikipedia

Due to this history, we can at least say that it doesn't appear to be a new symbol made up for $\hbar \equiv \frac{h}{2 \pi} ,$ but rather a preexisting symbol.


$\BackUp$ $\textbf{In 1900:} ~~ \textbf{Planck's constant,}~ `` h ", ~\textbf{appears.} \Target{In1900}$

In 1900, Max Planck came up with Planck's law, $$ {B}_{\nu} \left( \nu, T \right) ~=~ \frac{2 h {\nu}^{3}}{c^2} \frac{1}{{e}^{\frac{h \nu}{k_{\text{B}} T}} - 1} \,, $$ where

  • ${B}_{\nu} \left( \nu, T \right)$ is the spectral radiance of the black-body radiation;

  • $\nu$ is the frequency of emitted black-body radiation;

  • $T$ is the temperature of the black-body emitting the radiation;

  • $k_{\text{B}}$ is the Boltzmann constant;

  • $h$ is the Planck constant;

  • $c$ is the speed of light in the medium.

As a heuristically established law, it involved an unspecified value that came to be known as Planck's constant, $h .$


$\BackUp$ $\textbf{In 1913:}~~\textbf{The value}~{\frac{h}{2 \pi}}~\textbf{becomes notable.} \Target{In1913}$

In 1913, Niels Bohr proposed the Bohr model of the atom.

Bohr's model included stationary electron orbitals in which electrons had an angular momentum consistent with $$ m_{\text{electron}}vr ~=~ n \frac{h}{2\pi} \,,$$ where:

  • $m_{\text{electron}}$ is the mass of an electron;

  • $v$ is the orbital velocity of the electron;

  • $r$ is the radius of the electron's orbit;

  • $h$ is Planck's constant;

  • $\pi$ is the circle-constant;

  • $n \in \mathbb{N}$ is a non-zero, non-negative integer value.

This can be more concisely written as $$ m_{\text{electron}}vr ~=~ n \hbar \,,$$ such that there's now some motivation to have a symbol that's $\equiv \frac{h}{2 \pi} .$


$\BackUp$ $\textbf{In 1926:}~~\textbf{Papers define both }~{K \equiv \frac{h}{2 \pi}}~\textbf{and}~{\hbar \equiv \frac{h}{2 \pi}\,}\textbf{.} \Target{In1926}$

In 1926, both $K$ and $\hbar$ are defined as $\frac{h}{2 \pi} .$(Ref. 1)

  1. Erwin Schrödinger defined $K \equiv \frac{h}{2 \pi}$ in

    • Schrödinger, Ann. D. Phys., 79, 361-376 (1926).(Ref. 2)
  2. Paul Dirac defined $\hbar \equiv \frac{h}{2 \pi}$ in

    • Dirac, Proc. Roy. Soc., A112, 661-677 (1926).(Ref. 3)

Dirac's 1926 publication appears to be the first known, public use of $\hbar \equiv \frac{h}{2 \pi} ,$ though the paper itself introduces the symbol without explanation.


$\BackUp$ $\textbf{In 1930:}~~\textbf{Dirac again publishes}~ {\hbar \equiv \frac{h}{2 \pi}} ~ \textbf{in a book.} \Target{In1930}$

In 1930, Paul Dirac publishes a book, "The Principles of Quantum Mechanics", which defines $\hbar \equiv \frac{h}{2 \pi} ,$ as he did in his earlier 1926 paper.

As in his earlier 1926 paper, Dirac doesn't explain why the symbol $`` \hbar "$ was selected when defining it.


$\BackUp$ $\textbf{Conclusion:}~~\textbf{It's unclear exactly why}~ \mathbf{`` \hbar "} ~\textbf{was selected.} \Target{Conclusion}$

We can reasonably estimate that $\hbar \equiv \frac{h}{2 \pi}$ was selected by Paul Dirac (or someone close to him) at some point between 1913 (at which point the value became notable) and 1926 (at which point the definition was published).

I think it's a pretty safe bet that the symbol $`` \hbar "$ was selected in part due to its similarity to the symbol for Planck's constant, $`` h ".$ This seems like a perk over Schrödinger's contemporaneous $K \equiv \frac{h}{2 \pi} .$ $`` \hbar "$ probably got a boost over alternatives, e.g. $`` K " ,$ due to appearing in Dirac's influential book in 1930.

However, it's unclear why Paul Dirac may've chosen $`` \hbar "$ over some other variant of $`` h " .$

More information on the topic might come from an examination of Paul Dirac's personal notebooks or correspondence, though at the moment, the exact history seems unclear.


$\BackUp$ $\textbf{Errata} \Target{Errata}$

According to (Ref. 1), $`` \hbar "$ was introduced in Dirac's 1926 paper. (Ref. 1) claims to quote here $`` \hbar "$ appears, and they explicitly write in large, red text that Dirac's 1926 paper is where this notation came from.

However, looking at Dirac's 1926 paper, it seems that it actually redefines $`` h "$ as $\equiv \frac{h}{2 \pi} ,$ without using the symbol $`` \hbar " .$

Since this is an early paper with a special symbol, perhaps it's possible that other printings of the same paper used $`` \hbar "$ rather than $`` h " ,$ as claimed by (Ref. 1)? However, this could just be a misattribution on their part.

If this is just an error, then Dirac's 1930 book, "The Principles of Quantum Mechanics", would seem to be the next earliest sighting of $\hbar \equiv \frac{h}{2 \pi}$ found so far, assuming it actually appears like this in the first edition of the 1930-book. So far, I've just checked the third-edition, which wasn't published until 1947.

The above answer hasn't yet been corrected to account for this apparent error.


$\BackUp$ $\textbf{References} \Target{References}$

  1. $\Target{Ref1}$"The Planck constant h and the Dirac constant ħ. Their units and their history",
       by Ian Mills and P. R. Bunker.
       [PDF]

  2. $\Target{Ref2}$"Quantisierung als Eigenwertproblem" ("Quantization as eigenvalue problem"),
       by Erwin Schrödinger (1926)
       doi:10.1002/andp.19263851302

  3. $\Target{Ref3}$"On the theory of quantum mechanics",
       by Paul Adrien Maurice Dirac (1926-10-01).
       doi:10.1098/rspa.1926.0133.

  4. $\Target{Ref4}$"The Principles of Quantum Mechanics",
       by Paul Adrien Maurice Dirac (1930)

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  • $\begingroup$ That symbol for Saturn is a bit different. But it is the same as the alchemical symbol for lead. (The seven planets and the seven metals correspond, of course.) $\endgroup$ – Gerald Edgar Sep 15 at 9:50
  • $\begingroup$ @GeraldEdgar Added the alchemical symbol for lead! As for subtle differences, I added a qualifier to loosen it. Before posting this answer, I did look around at some different versions of the astronomical symbol, and they seem to show some variance; for example, the first image from Wikipedia shows a table from a publication in 1850 that has it looking a lot like a (non-italicized) h-bar. I figure that a lot of the formalization of slight differences is probably retro-historical. $\endgroup$ – Nat Sep 15 at 10:52
  • $\begingroup$ @Nat, You did a really thorough search. I feel the h-bar symbol must have Latin origins, because Erwin and Dirac must have studied Latin in school (I assume). By the way, I am looking the origin of sinc function. the detail question is here: mathoverflow.net/questions/341436/… $\endgroup$ – M. Farooq Sep 15 at 15:01
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    $\begingroup$ I do not think SE MathJax supports section labels, see Using labels with mathJax? You can accept your self-answer by clicking on the checkmark. $\endgroup$ – Conifold Sep 29 at 4:56
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    $\begingroup$ Vow. Maybe you can add to that thread. I couldn't figure it out reading it, and still can't looking at your code :) I wish I could upvote your answer again. If something new comes up it can be added, but it's just that people are more likely to read a thread with an accepted answer. And speaking from experience, most answers on hsm are far less comprehensive than yours, accepted ones included. $\endgroup$ – Conifold Sep 29 at 8:30
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There is another myth that h is a short form of Hilfsgrösse, with no proof whatsoever (see the excerpt below). Thus "h-bar" is no different myth, no matter how reliable it sounds. A very valid question is who introduced the h-bar notation. Since h-bar is also called Dirac h, I checked his book, and indeed there it is on page 87, of his famous book "Principles of Quantum Mechanics"

"$uv-vu$=$\hbar$$i$[u,v], where $\hbar$ is a new universal constant. It has the dimensions of action. In order that theory may agree with experiment, we must take $\hbar$ equal to $h$/2$\pi$, where $h$ is the universal constant that was introduced by Planck, known as Planck's constant."

Have a look at this anecdote The Thermal Radiation Formula of Planck (1900)

Myth about symbols

Long time ago we were having a chemistry exam in school. A student said what if there is question, "why a beaker is called a beaker?". In all seriousness another one quipped that "a beaker is a beaker because it has a beak." I was impressed, thinking that indeed the beaker's spout looks like a bird's beak and thought this is the right answer. When I came home and checked the dictionary, this cute story had nothing to do with reality. Don't trust whatever you find on the web. A prime example is fake mentioned anecdote, just like I discovered yesterday that nobody knows who coined sinc function's full name. Books, webpages, all say it is sinus cardinalis or cardinal sine. It may be, but the whoever came up with this full name is not known and all wrong names are associated with it.

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    $\begingroup$ Good find on the Dirac book! Looks like Dirac's the alleged original source of the symbol, first published in a 1926 paper, "On the Theory of Quantum Mechanics", Dirac, Proc. Roy. Soc., A112, 661-677 (1926), right below Eq. (1) on printed-page 661. Unfortunately, no explanation for the choice of $`` \hbar "$ there, either. $\endgroup$ – Nat Sep 14 at 2:21
  • $\begingroup$ It means there is no explanation because the inventor never told us. The rest is all speculation. $\endgroup$ – M. Farooq Sep 14 at 2:24
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    $\begingroup$ Yeah, unfortunately it may not be an answerable question. Still, maybe one of those archive projects might scan some of Dirac's personal notebooks, which may show where he, personally, started to use it in his own work? I know in my own works, I usually have reasons for selections, though I don't always share them as it'd be too cumbersome. Still, if someone were to trace back through my journals, they could watch my notation and terminology evolve over time, back to when I first started writing terms. It'd be cool if Dirac's own notebooks are so revealing, and perhaps available somewhere? $\endgroup$ – Nat Sep 14 at 2:27
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Dirac was not free to create a new symbol, because publishing would be prohibitively expensive due to printing costs. So the choice was limited to existing symbols. Many printers probably had the IPA-symbols, as it was used in dictionaries. Around 1930, h-bar had been added to IPA. (link)

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    $\begingroup$ Generally "I guess" answers are worthless here. References are needed. $\endgroup$ – Gerald Edgar Sep 26 at 1:00
  • $\begingroup$ Ok, I removed the offensive word "guess". $\endgroup$ – jkien Sep 29 at 8:06

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