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My question is very naive and could sound strange but it seems to me natural in so far as the Planck constant is related to the first quantization (of newtonian particle mechanics/galilean relativity) while the Dirac one shows up in the second quantization (of fields theory/special relativity)... Anyway I think the conceptual difference between the two constants is rarely emphasized.

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up vote 6 down vote accepted

"First quantization" and "second quantization" are widely used names for the same procedure applied to two different classical systems – classical mechanics and classical field theory. In both cases, the operation introduces Planck's constant and it's always the same constant. For example, the angular momentum $J_z$ is a multiple of $\hbar/2=h/4\pi$ both in non-relativistic quantum mechanics for a few particles as well as quantum field theory – this fact results both from "first quantization" as well as "second quantization". It is a universal fact of quantum mechanics. In the same way, Schrödinger's equation contains the factor of $\hbar$ and this equation holds for non-relativistic quantum mechanics as well as quantum field theory (with an appropriate Hamiltonian that encodes the total energy).

The reason why $\hbar$ is more often found in quantum field theory and $h$ is more often found in simpler discussions of quantum mechanics is that $h$ is associated with frequency $f$ which is the quantity chosen by physics beginners while the advanced physicists usually consider the angular frequency $\omega=2\pi f$ to be more natural, and that's why they also talk about $\hbar$. Note that the energy of a photon (or another quantum) is $E=hf=\hbar\omega$; the factors of $2\pi$ cancel.

The constant $\hbar$ is more fundamental because it naturally appears in the Heisenberg equations, Schrödinger's equations, Feynman's path integral, commutators of $x,p$, and so on. These are the fundamental equations and it would be a waste of time to write an extra $1/2\pi$ in all of them. A physics beginner doesn't really understand these fundamental equations well. He prefers to look at things like $E=hf$ which may be written in a simple way using $h$ as long as we express the frequency by $f$.

But $\hbar=h/2\pi$ always holds – they're not independent at all. They're just two constants associated with two conventions and the more one knows about quantum theories, the more likely he is to switch to $\hbar$.

Let me also mention that the Dirac constant is $\hbar$ and not $\hbar/2$.

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Thank you Lubos! I understand the fundamental significance of angular frequency ω in advanced physics as you explain it and I know its relevance in dimensional analysis too : I have in mind the argument to obtain (modulo a constant factor close to one) the correct electromagnetic (2/3) or gravitational (32/5) power radiated under the assumption of a dipolar (ω^4) or quadrupolar ( ω^6) emission ! Now coming back to this half integer spin or Dirac operator that is some kind of a square root of the Laplacian I can't help thinking that there is more than one mystery in quantization ... – laboussoleestmonpays Feb 26 '13 at 20:38

Nah, they're the same. Even Planck's constant comes from fields; he was looking at the electromagnetic radiation (which is all field, all the time) getting kicked off by a warm black body.

Also, while most people use $h \nu$ for the energy of a photon, in grad school (physics) we often used $\hbar \omega$. Also, usually I've seen the Dirac constant defined as $\hbar$, rather than $\frac{\hbar}{2}$. We often use the term Planck's Constant for both interchangeably (Don't worry, Dirac gets his due in physics). The first time I learned the Schroedinger Equation, it was with $\hbar$ (I now have it tattooed across my back in the same form). Here's a picture of the tattoo. Notice the Dirac, or "reduced Planck's" Constant

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Cheat fail: when you're in an exam and you forget the Schr. eqn, all you can do is help the person sat behind you. – twistor59 Jan 19 '13 at 16:47
Ha ha ha, I am laughing aloud! This was completely unexpected. This is the weirdest answer I have seen here. Upvoted! (+1) – Eduardo Guerras Valera Jan 20 '13 at 7:11
Thank you for your answer "thatnerd"! I know that Dirac got his due in physics but since we owe him the "discovery" or unveiling of spinors in the real physical world I regret that the Dirac constant is ℏ and not ℏ/2 ... – laboussoleestmonpays Feb 26 '13 at 19:42

The factor $\frac{1}{2\pi}$ it is a matter of convenience. Dirac constant, or Planck's reduced constant can show up and indeed it does in quantum statistical mechanics. If the inverse of $2\pi$ is going to appear a lot of times,you should use $\hbar$ instead of $h$ but, as far as I know, it has nothing to do with your proposal.

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The fact that $h$ appears in first quantization and $\hbar$ is purely a matter of convention. Since $$\hbar = \frac{h}{2 \pi} $$ we may as well write equations like $$E = h \nu $$ as $$E = \hbar \omega $$ and in fact that is a frequent form of that equation.

Multiplication and division by a constant doesn't make the two independent of each other. The physical content is still exactly the same.

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