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This question is related to Planck units (also called natural units, absolute units or God's units). I'm wondering which constant is the most fundamental and should be normalized to 1. I would like as much explanation as possible please.

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Generally $\hbar$ is normalised to $1$ There is no reason why you cannot normalise $h$, but $\hbar$ is the convention for similar reasons to why you define $\hbar=\frac{h}{2\pi}$ in the first place; it gets rid of lots of annoying factors of $2\pi$. Here is a wikipedia article on the subject, which rightly points out that there are a number of different normailsation systems you could use. This is just the most widely used one.

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I would say that $\hbar$ is the more fundamental constant, because it has a mechanical interpretation: it is the size of one "lump" of angular momentum. We have $h$ because quantization of angular momentum wasn't understood for the first couple decades of the 20th century; $h$ converts between frequency (in particular units) and energy (in particular units). But you don't need a particular system of units to measure $\hbar$.

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    $\begingroup$ just to be clear: $\hbar$ converts between energy and angular frequency - for regular frequence, you need regular $h$ $\endgroup$ – Christoph May 15 '14 at 15:13
  • $\begingroup$ @Christoph Sure it does. My point is that I'm not aware of a mechanical phenomenon whose value is $h$, while $\hbar$ as the quantum of spinning is absolutely fundamental to quantum mechanics. $\endgroup$ – rob May 15 '14 at 15:16
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    $\begingroup$ So what you're saying is that $h$ is related to ordinary frequency (in cycles per second or hertz), while $\hbar$ is related to angular frequency (in radians per second)... But why exactly would angular frequency be more fundamental than ordinary frequency? $\endgroup$ – Quantum Force May 15 '14 at 15:37
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    $\begingroup$ And yes I just checked, you're right, $\hbar$ is the quantum of angular momentum, that makes $\hbar$ something very fundamental for sure... But $h$ is also the quantum of something: $h$ is the quantum of action. So why would angular momentum be more fundamental than action? Or why would action be more fundamental than angular momentum? $\endgroup$ – Quantum Force May 15 '14 at 15:38
  • $\begingroup$ @QuantumForce The action is a very useful device for solving equations of motion. But does the action have a physical reality the way that angular momentum does? I'm not sure. $\endgroup$ – rob May 15 '14 at 16:06
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The question has two aspects:

  1. Which constant is more fundamental?
  2. Which constant should be set to 1?

@rob and @nivag have given excellent answers to the question in the title, and those answers are deserving of upvotes. While their answers focus on Aspect #1 above, I'd like to focus on Aspect #2.

Why do you want to normalize a constant to 1 in the first place?

As a physicist myself, I am familiar with doing this procedure and know that there are several reasons you may wish to do this. All properly defined unit systems are consistent with each other, so the decision to normalize any constant to 1 depends on what the unit system is designed for. There may be a pragmatic or a theoretical reason for doing this.

Somebody working with spectra may wish to work in units where $h=1$, while a student of quantum mechanics may wish to define $\hbar = 1$. This is the pragmatic reason, wherein you just want to get rid of pesky constants. On the other hand, somebody trying to develop a unifying theory of everything may wish to develop a system of units where the "most fundamental" constant is set to 1. This is the theoretical reason, wherein you're seeking a "proper" or "elegant" representation, though extra factors may be introduced.

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