M1-intro-note.txt

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With this short note about-- I've talked about resonances.
I've talked about, now, the two important parameters,
the resonance frequency and the full width at half maximum.
How do we measure those?
And there is actually sometimes a confusion.
The more systematic approach is you
should measure all those frequency and line
widths in angular frequency units, which
are technically radian per second, 2 pi per second.
But since radian has no dimension,
we sometimes say we measure it in inverse seconds.

So this is a measurement of angular frequencies,
and this is different from the unit of frequencies.
When we have a frequency, which is an angular frequency divided
by 2 pi, frequencies are usually measured in hertz.
The problem is that hertz is always also 1 over a second,
and this is where the confusion comes.
So let me just point out how you can avoid the confusion.
You may write an angular frequency omega 0.
It is 2 pi times 1 megahertz.
Then you exactly know what it is.
You can, of course, this is nothing else than 6 times--
6.28 times 10 to the sixth second to the minus 1,
but you should never say that omega 0 is
6.8 times 10 to the 6 hertz.
Because then people don't really know, and you get confused.
And you confuse other people, if you really
mean that this now has a frequency of 6 times 10
to the 6 hertz or an angular frequency.
So just be clean in your thinking, in your homework,
and all that, that a frequency, when you mean angular
frequency, is 1 over second.
When you mean it as a frequency, it's hertz.
And this is often the clearest form to say,
yes, I know where to put the 2 pi and I put it in explicitly.
So we often, in our papers, report frequencies like that.

Finally, there is a question about gamma.
So what are the units for gamma?
Well, if you look at the exponential, which decays,
it has e to the minus i omega t, and then it
has the imaginary part gamma t.
So gamma is really a temporal decay,
and there is no question about frequency and angular
frequency.
It's not a frequency.
It's not an angular frequency.
It's a decay rate.
So for instance, if gamma is 10 to the 4 per second,
you should never say gamma is 10 to the 4 hertz,
or you should also never say gamma is 2 pi times
1.66 kilohertz.
That just doesn't make any sense.
Gamma is really a damping rate and is therefore
an inverse time.
The damping time associated with this gamma
is simply the inverse of it, and in the case chosen,
it's 100 microseconds.