M1L1a.txt

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We want to start talking about a seemingly simple, but very
complex system in physics-- the harmonic oscillator.
So the next part is actually due to Professor Vladan Vuletic,
who worked out the topic very nicely about how precisely can
you measure frequencies.
And I don't need to remind you that some
of the most accurate measurements in all of physics
are done by measuring frequency.
It's actually a kind of unwritten rule.
If you want to measure something precisely,
make sure you find a way that this quantity can be measured
in a frequency measurement.
Because frequencies, that's what we
can measure, with synthesizers, with clocks and such.
So therefore, the question how precisely can you
measure frequency is a question which
is actually relevant for all precision measurements.
So well, we know we have Fourier's theorem.
If you have an oscillator which oscillates for a time, delta t,
we have the Fourier theorem, which
says we have a finite width of the frequency
spectrum, delta omega.
But there is a spread of frequency components involved
in such a way that delta omega times delta t
is larger or equal than 1/2.
The case of 1/2 is realized for Gaussian wave packets.
And of course, this Fourier limit
should remind you of Heisenberg's uncertainty
relation.
Of course, Fourier limit and Heisenberg uncertainty relation
are related, because what Heisenberg expressed
turns out to be simply the limit due to the wave
nature of matter.