Hi when you say quantum mechanics to
someone, one of the things they first
think of is usually the Heisenberg's
Uncertainty Principle. But it's also one
of the concepts that people
misunderstand the most.
Now in quantum mechanics remember
everything's a wave, and so it doesn't
have one perfect position unless all the
wave is piled up into one place, and also
because different parts of the wave can
move around, it doesn't necessarily have
one perfectly well defined velocity or
momentum either. And it turns out that if
you multiply the spread in mentem the
variance the momentum times the spread
in position either variance in position
if you multiply those two variances
together then those can't get smaller
than h-bar on two so what that means is
you can have a way there's a really
well-defined momentum but if you do then
it must be really spread out conversely
if you have something that has a really
well-defined position that it must have
a very large spread momentum and so if
you look at what happens next it's going
to blow up and the way people talk about
this and indeed the way Heisenberg
originally talked about this is in the
context of trying to measure something
so Heisenberg talked about trying to
measure something microscopic if you try
and measure something you have to
interact with it in some way if you
don't ever interact with something you
never get any kind of idea of how it's
behaving and so he talked about cases
where if you're trying to measure the
position of something then you might
interact throw light at it in some way
and the light bouncing off it will
change its momentum it will give it a
kick and that's true but the problem
with that approach is that it tends to
make you continue to try and think of
clever ways you might try and measure
something without giving it kicks
whereas in fact the Heisenberg
uncertainty principle is not about how
hard it is to measure something but
about the properties that the thing can
have in and of itself so remember that
everything in quantum mechanics is
described by a wave whether it be light
or matter or anything else and there's a
relationship between the momentum of
that wave and its wavelength in other
words the actual wave that has a
perfectly ball to fly momentum has a
perfectly well defined wavelength in
other words it's a perfect sinusoidal
sign with some phase going along in
space and so we have a perfectly well
defined wavelength here so something
that has a perfectly well defy momentum
has an uncertain even mention that's
essentially zero they can also see that
this wave doesn't ever dip down so it's
just as likely to be a million miles
that way is is to be a million miles
that way so the uncertainty in position
is effectively infinite
and so we can make one of these
quantities very small but other than
costs of making the other one very very
large now contrast that to the wave that
has a very well-defined position so it's
0 for most of space and then it has some
finite value and then is mostly 0
everywhere else and so you can see that
this wave function has a fairly well
defined position and so its uncertainty
in position might be something like that
being and so this number here is very
small unfortunately it doesn't now have
a well-defined wavelength in order to
make this wave here you have to add a
lot of different kinds of sine waves
together so you get a lot of different
kind of mentor and so it has a large
spread my mentor so get a small spread
in position and we get a large spread
mentor and so this is trade-off between
the spread in position and the spread of
momentum and so another way of writing
Heisenberg's uncertainty principle Zyzz
say a way that has a well-defined
momentum just doesn't look like a wave
that has a well-defined position so this
equation here is a graphical version of
Heisenberg's uncertainty principle now
it's not just momentum and position that
are related in this way in quantum
mechanics there are lots of quantities
that have an uncertainty principle
relationship between them and one of the
more commonly quoted ones is actually
energy in time in terms of the
underlying mathematics of quantum
mechanics this is a slightly unusual one
because why you can talk about how
spread out a wave is it's pretty hard to
talk about how spread out the time or
wave is in the absence of the object
being described by this way of being
created or destroyed then the time that
exists is essentially all time forever
and so what exactly is this equation
trying to tell us well in quantum
mechanics the energy of a wave
determines how it changes so in
Schrodinger's equation the rate of
change of the wave function is
proportional to the energy of that wave
function and so if you have a wave with
one perfectly well defined energy it
might still have a spread in momentum it
might still have a spread in position
but it doesn't change an example of that
might be something sitting in absolute
zero it might be an atom City in its
ground state and so you have a single
wave that static for all time in order
to have something changed so for changes
to happen
so a measurement energy transfer or
something like that the energy
difference the spread of energies in
that way if defines how fast that
process goes and so there's a
relationship between the spread and
energies and the time scale of any
process
