What Determines the Use of Normal vs Angular Frequency in Quantum Transitions?

[kelly0303]

This might be silly, but I am a bit confused by the conventions used for transitions in atoms/molecules. Usually these are given in nm and using the formula $\lambda = c/\nu$, from here we can get the normal frequency $\nu$ and the angular frequency would be given by $\omega = 2\pi\nu$. In all the conversion apps I found, when switching from nm to Hz, they give the value consistent with $\nu$, not with $\omega$. So if we measure the energy splitting using $\nu$, this implies we assume that the Plank constant is $h=1$ (and not $\hbar = 1$), right? So what is used depends solely on whether we set $h=1$ or $\hbar=1$? Based on what I said above, it seems like in tables with values of transitions (and conversion between units) they use $h=1$, but in most textbooks they assume $\hbar=1$. Am I missing something?

Then when talking about a sinusoidal electric field frequency (in that case it is an actual frequency, not energy), the formula is $e^{i\omega_E t}$. So here, in order to be on resonance with a transition, given that we use $\nu$ and not $\omega$ for the transition splitting, it is implied that we want $\nu = \frac{\omega_E}{2\pi}$?

Is there an easy way to keep track of these? Am I overthinking the notation? Thank you!

 

[osilmag]

Where is it that you find Planck's Constant equal to one?

 

[kelly0303]

Where is it that you find Planck's Constant equal to one?

I actually didn't find it explicitly, but this is what confuses me. For example, this paper (nothing special about it, but it helps me explain my point), in Figure 2 (and everywhere in the paper) they use MHz. They don't mention anything about their convention for the Plank constant. Now, we have that $E = h\nu = \hbar \omega$. So if you quote the energy in MHz, if you assume $h=1$ then the quoted value is $\nu$, if you assume $\hbar=1$ then the quoted value is $\omega$. So if I assume that $\hbar=1$, this means that if I want the frequency of the transition, I need to divide everything in that paper by $2\pi$.

On the other hand, on a units conversion page like this, if you for example convert 1 nm to GHz, you get 299,792,458 GHz, which is the value you'd get for $\nu$, not $\omega$. So this means they assume $h=1$, no? Also this is consistent with all the unit conversion sites I found.

Basically my confusion is, if I get a frequency from a paper in, say, MHz and I want to convert it to some other units, do I always need to divide the value from the paper by $2\pi$ before doing the conversion i.e. do I always have to assume the paper uses $\hbar=1$ unless otherwise specified? Also, why would these websites use $h=1$ if most of the results in the papers assume $\hbar=1$?

 

[mjc123]

Hertz is, properly speaking, cycles per second, i.e. frequency ν. Angular frequency ω is radians per second. Both have the dimension s^-1, which can be confusing.ν But if someone is talking MHz, yiou have to assume they are talking about ν.

 

[osilmag]

If you assume that Planck's Constant is one, then the energy is at the same magnitude as the frequency. I don't think that is accurate.

$F = \omega / 2 \pi$. Yes, you would divide $\omega$ by 2$\pi$ to get frequency in Hz.

Fixed conversion between frequency in Hertz and radians

 

[osilmag]

Fixed conversion between frequency and radians

 

[kelly0303]

Hertz is, properly speaking, cycles per second, i.e. frequency ν. Angular frequency ω is radians per second. Both have the dimension s^-1, which can be confusing.ν But if someone is talking MHz, yiou have to assume they are talking about ν.

This is what I am thinking, too. But doesn't this assumes $h=1$, while the usual convention is $\hbar=1$?

 

[kelly0303]

Fixed conversion between frequency and radians

What do you mean by this? I know how to convert between the 2, but when someone says in a paper that the frequency is 100 MHz, without specifying their convention, should I assume this is a frequency i.e. $h=1$ or an angular frequency i.e. $\hbar=1$. This makes a huge difference in practice if I end up being off by a factor of $2\pi$.

 

[hutchphd]

What do you mean by this? I know how to convert between the 2, but when someone says in a paper that the frequency is 100 MHz

Cease.
You are conflating things that need not be and should not be conflated.
The issue of angular measure is usually not a problem. The natural unit is radians but many others exist and should be specified. cycles is cycles (1) Degrees is degrees (360/cycle)
If someone says the frequency is 100 MHz why are you unable to convert this to energy without
ambiguity? Plancks constant is uniquely defined constant, written in several forms
If you are really still confused just always use radian measure until confusion passes.

 

[kelly0303]

Cease.
You are conflating things that need not be and should not be conflated.
The issue of angular measure is usually not a problem. The natural unit is radians but many others exist and should be specified. cycles is cycles (1) Degrees is degrees (360/cycle)
If someone says the frequency is 100 MHz why are you unable to convert this to energy without
ambiguity? Plancks constant is uniquely defined constant, written in several forms
If you are really still confused just always use radian measure until confusion passes.

I don't think you understand my confusion. My question is not about a self contained paper, it is when I have to compare different papers or compare atomic frequencies with lasers. For example, if I read in a paper that a transition is at 300 THz and I want to drive that transition with a laser, what wavelength do I use in the laser: 999.3 nm or 159.0 ($999.3/(2\pi)$) nm and why (assuming the paper doesn't make any mentions about they conventions)?

 

[hutchphd]

No mention is necessary. The customary expression $$c=\lambda f $$ where f is in Hz. Always.
An alternative equivalent expression is $$c= \frac \omega k $$ where frequency is in radians/second and k is the wavenumber defined by $$k=\frac {2\pi} \lambda$$ Always
Perhaps the issue is lack of care in specification of wavenumber. Without notation, in my experience the units are radians/length.
Do you have an ambiguous example?