Hi everybody. I have decided that this channel
lacks a history part because there is so much
we can learn from the history of science.
So today I want to tell you a story. It’s
the story of how Werner Heisenberg got the
uncertainty principle named after him.
Heisenberg was born in 1901 in the German
city of Würzburg. He went on to study physics
in Munich. In 1923, Heisenberg was scheduled
for his final oral examination to obtain his
doctorate. He passed mathematics, theoretical
physics, and astronomy just fine, but then
he run into trouble with experimental physics.
His examination in experimental physics was
by Wilhelm Wien. That’s the guy who has
Wien’s law named after him. Wien, as an
experimentalist, had required that Heisenberg
did a “Praktikum” which is a series of
exercises in physics experimentation; it’s
lab work for beginners, basically. But the
university lacked some equipment and Heisenberg
was not interested enough to find out where
to get it. So he just moved on to other things
without looking much into the experiments
he was supposed to do. That, as it turned
out, was not a good idea.
When Heisenberg’s day of the experimental
exam came, it did not go well. In their book
“The Historical Development of Quantum Theory”,
Mehra and Rechenberg recount:
“Wien was annoyed when he learned in the
examination that Heisenberg had done so little
in the experimental exercise given to him.
He then began to ask [Heisenberg] questions
to gauge his familiarity with the experimental
setup; for instance, he wanted to know what
the resolving power of the Fabry-Perot interferometer
was... Wien had explained all this in one
of his lectures on optics; besides, Heisenberg
was supposed to study it anyway... But he
had not done so and now tried to figure it
out unsuccessfully in the short time available
during the examination. Wien... asked about
the resolving power of a microscope; Heisenberg
did not know that either. Wien questioned
him about the resolving power of telescopes,
which [Heisenberg] also did not know.”
What happened next? Well, Wien wanted to fail
Heisenberg, but the theoretical physicist
Arnold Sommerfeld came to Heisenberg’s help.
Heisenberg had excelled in the exam on theoretical
physics, and so Sommerfeld put in a strong
word in favor of giving Heisenberg his PhD.
With that, Heisenberg passed the doctoral
examination, though he got the lowest possible
grade.
But this was not the end of the story. Heisenberg
was so embarrassed about his miserable performance
that he sat down to learn everything about
telescopes and microscopes that he could find.
This was in the early days of quantum mechanics
and it led him to wonder if there is a fundamental
limit to how well one can resolve structures
with a microscope. He went about formulating
a thought experiment which is now known as
“Heisenberg’s Microscope.”
This thought experiment was about measuring
a single electron, something which was actually
not possible at the time. The smallest distance
you can resolve with a microscope, let us
call this delta x, depends on both the wave-length
of the light that you use, I will call that
lambda, and the opening angle of the microscope,
epsilon. The smallest resolvable distance
is proportional to the wave-length, so a smaller
wave-length allows you to resolve smaller
structures. And it is inversely proportional
to the sine of the opening angle. A smaller
opening angle makes the resolution worse.
But, said Heisenberg, if light is made of
particles, that’s the photons, and I try
to measure the position of an electron with
light, then the photons will kick the electron.
But you need some opening angle for the microscope
to work, which means you don’t know exactly
where the photon is coming from. Therefore,
the act of measuring the position of the electron
with a photon actually makes me less certain
about where the electron is because I didn’t
know where the photon came from.
Heisenberg estimated that the momentum that
would be transferred from the photon to the
electron to is proportional to the energy
of the photon, which means inversely proportional
to the wavelength, and proportional to the
sine of the opening angle. So if we call that
momentum Delta p we have Delta p is proportional
to sine epsilon over lambda. And the constant
in front of this is Planck’s constant, because
that gives you the relation between the energy
and the wave-length of the photon.
Now you can see that if you multiply the two
uncertainties, the one in position and the
one in momentum of the electron, you find
that it’s just Planck’s constant. This
is Heisenberg’s famous uncertainty principle.
The more you know about the position of the
particle, the less you know about the momentum
and the other way round.
We know today that Heisenberg’s argument
for microscopes is not quite correct but,
remarkably enough, the conclusion is correct.
Indeed, this uncertainty has nothing to do
with microscopes in particular. Heisenberg’s
uncertainty is far more than that: It’s
a general property of nature. And it does
not only hold for position and momenta but
for many other pairs of quantities.
Many years later Heisenberg wrote about his
insight: “So one might even assume, that
in the work on the gamma-ray microscope and
the uncertainty relation I used the knowledge
which I had acquired by this poor examination.”
I like this story because it tells us that
if there is something you don’t understand,
then don’t be ashamed and run away from
it, but dig into it. Maybe you will find that
no one really understands it and leave your
mark in science.
Thanks for watching, see you next week.
