Hi guys, how are you? It is lovely to back
with some brand new stuff; this time we’ll
be covering the very foundations of quantum
mechanics. Isn’t that super exciting? To
start with – since there is so much misinformation
surrounding this topic - I decided to make
a whole new series covering Heisenberg’s
Uncertainty Principle. This is a series for
everybody, from beginner's to more advanced
levels – no matter how little or how much
you already know, I am hoping you’ll find
something that catches your interest. You
are now watching the first video in the series;
we’ll start with a little home experiment,
we’ll then cover some theory and finally,
we’ll finish with a few very interesting
examples. I hope you enjoy it… let’s get
started!
Let’s start with a little experiment to
demonstrate Heisenberg’s Uncertainty Principle
which can be easily set up at home and it
is cheap and fun to do. We are going to need
a laser pointer, a couple of very sharp blades,
a clothes peg and some playdough. In addition,
we will also need a smooth wall to act as
our projector screen, where the laser will
be fired.
Ok, so here’s the setup. As you can see,
the playdough is being used to hold the laser
pointer in place and also to hold the two
blades in place. It will allow us to move
the blades so that the distance between them
can be modified at will. The clothes peg is
simply being used to press the “on” button
down so we don’t have to use our hands.
Now, once we have pointed the light beam right
in the middle between the two blades, we can
then proceed to narrow the slit’s width
by slowly bringing the two blades towards
each other. Be extra careful because those
blades are very sharp and I’m guessing you
don’t want to end up injuring yourself.
So - unlike what my son is doing in this clip
- try your best to keep the blades as parallel
to each other as you possibly can while very
slowly bringing them closer together. Now,
let’s pay attention to what happens to the
laser spot on the wall as we make the distance
between the blades narrower and narrower.
At first – as expected – the red dot gets
smaller as we make the slit smaller. However,
at a certain point something remarkable happens.
As we make the slit even narrower, the spot
begins to spread out, it starts to get wider
and wider. In addition, some very interesting
fringes start to appear. What is going on
here?
As you may know, the wave nature of light
- in particular the phenomenon of interference
- is the reason why these series of light
and dark fringes are created. However, the
quantum mechanics behind the creation of these
fringes is material for another series altogether.
Here we are going to focus on how the Uncertainty
principle can explain the counter-intuitive
fact that the area where the photons land
becomes wider and wider as we narrow the slit’s
width past a certain point. But first, a little
bit of theory.
Alright… so now that we’ve seen the uncertainty
principle in action, let’s find out what
it actually states. Heisenberg’s Uncertainty
Principle asserts that:
“In any system, in any state, the uncertainties
of position and momentum obey the following
inequality...”
...which states that the product of the uncertainty
in position and the uncertainty in momentum
is always larger than or equal to this other
quantity, h-bar over two, where h is Planck’s
constant.
Ok, so how can we rephrase this so that it
can be understood in a more intuitive way?
Well, let’s say the system in question is
a particle for instance – like an electron
or a photon - then the uncertainty principle
can be re-stated like this:
“It is impossible to find a state in which
the particle has definite values for both
position and momentum”
Two points to clarify here: one, momentum
(symbolised by the letter p) is equal to mass
times velocity, and since velocity has a direction,
so does momentum. Which brings me to my second
point: position and momentum components need
to be along the same directional axis for
the uncertainty principle to apply. See that
little x next to the momentum symbol? It basically
means that - in this case - we are talking
about momentum along the x direction. And
since the uncertainty principle applies to
any possible directional axes, we can write
with equal confidence the following inequalities
for the y and z axes...
Question: Does Heisenberg’s Uncertainty
principle apply to the following relationships?
The answer is absolutely not. There is no
constrain on pairs of position and momentum
components along different directions. Their
product doesn’t obey any inequality and
can, in principle, be equal to zero.
So, what does the uncertainty principle actually
tell us? It tells us that it is impossible
- not just hard or difficult – but that
it is impossible to find a quantum state in
which the particle has well-defined values
for both position and momentum (and remember,
we are talking about components along the
same axes here). Put another way, because
there doesn’t exist a state with definite
values for both position and momentum, there
is a fundamental limit to the precision with
which the position and the momentum of a particle
can be simultaneously known. We can’t know
them both at the same time for the simple
reason that well-defined values of both variables
cannot exist simultaneously.
The uncertainty principle is a result of profound
importance in quantum mechanics. Let me emphasise
something: contrary to popular belief and
to what some books or teachers might tell
you, Heisenberg’s Uncertainty Principle
is not the result of disturbances created
through the process of measurement, neither
it is a limitation resulting from poor instrumental
precision, nor it is a reflection of our failure
to gain knowledge about how Nature works.
No, none of that is true. The uncertainty
principle is in fact a result which can be
derived theoretically from the fundamental
principles of quantum mechanics, principles
which describe an inherent indeterminacy in
the relationship between certain pairs of
complementary variables, such as position
and momentum. The uncertainty principle is
inherent in the properties of wave-like systems,
and since quantum mechanics describes quantum
entities as probabilistic waves, indeterminacy
naturally arises. More on this later in another
video.
Now, let’s take a look at the Uncertainty
Principle through a couple of examples, starting
with the single-slit laser experiment I showed
you at the start of the video.
This is a diagram of the single-slit diffraction
experiment, which I demonstrated earlier on.
The photons in the laser beam have a momentum
of magnitude p moving in the x direction,
that is, perpendicularly to the slit, which
has a width w measured in the y direction.
Now, let’s consider one single photon. Whereas
in front of the slit the photon has definite
momentum p in the x direction (and hence zero
momentum in the y direction), behind the slit
the situation completely changes. The restriction
imposed by the slit means that the photon’s
position is now known to an accuracy of at
least w in the y-direction. Hence the width
of the slit - w - may be taken as the photon’s
position uncertainty in the y direction, that
is, we can write deltay=w.
Applying Heisenberg’s uncertainty principle
now – remember, we need to apply it in the
same direction, in this case, the y axis – we
get a non-zero momentum uncertainty, deltap
= h/(2w) , which means that - from behind
the slit onwards - the photon’s momentum
may end up having a non-zero component in
the transversal direction. And this is why
– for a sufficiently narrow slit – as
we look at the wall, we may find the photon
appearing somewhere further to the right or
to the left of where we would intuitively
expect it to appear. As we narrow the slit
more and more, we realise that the more we
try to localise the photon (that is, the more
we try to define its position) by shrinking
the slit’s width w, the more spread (the
more uncertainty) there is in its transversal
momentum.
When we consider all the photons present in
the laser beam, this spread in the uncertainty
of each of their momenta manifests as an angular
spread in the beam behind the slit, a broadening
of the diffraction pattern, which is precisely
what we saw in our experiment!
Here’s another interesting example. Heisenberg’s
uncertainty principle not only helps to predict
the approximate size of the hydrogen atom,
but it also explains why atoms are stable.
According to classical ideas based exclusively
on Maxwell’s equations of electromagnetism,
the negatively charged electron should continuously
emit electromagnetic radiation, since it is
constantly accelerating in its circular orbit.
The electron should, therefore, spiral towards
the nucleus of the atom. However, Heisenberg’s
uncertainty principle prevents this from happening.
Why?
Well, if the electron were to spiral in towards
the nucleus, its position would become more
and more precisely known, to the point where,
if the electron was really close to the the
nucleus, its position would be known with
such a precision that - by Heisenberg’s
Uncertainty Principle - the uncertainty in
the electron’s momentum would have to increase
by a massive factor, so massive that the electron’s
kinetic energy would increase by a factor
of around 10 to the power of 10. There is
no way for the electron to gain this amount
of kinetic energy; it is impossible since
its electrostatic potential energy is nowhere
near such high values, and by conservation
of energy therefore, this cannot happen. And
this is how Heisenberg’s Uncertainty principle
explains why the hydrogen atom is the size
it is and the fact that it is stable.
You may wonder whether quantum mechanics applies
just to the world of tiny things. The answer
is definitely not: quantum mechanics - including
the uncertainty principle - applies to everything,
no matter what size. The reason why we do
not usually observe the uncertainty principle
in the macro-world of everyday objects is
because of Planck’s constant, h, which is
incredibly small; so minute that the indeterminacy
imposed by the uncertainty principle becomes
very significant in the realm of atomic and
subatomic physics but it rarely becomes apparent
in everyday life.
Having said that, there are indeed very interesting
cases where the strange world of quantum mechanics
does manifest in an obvious way in macroscopic
objects. For instance, have you ever heard
of Bose-Einstein condensates? A Bose-Einstein
condensate is a state of matter obtained by
cooling down a gas of atoms of extremely low
density to a very, very low temperature. In
order to keep the sample cool, the gas is
trapped in a small volume of empty space by
magnetic fields and beams of light; this keeps
it away from the relatively warm walls of
the container. This fascinating state of matter
was first predicted by Albert Einstein and
Satyendra Nath Bose in 1925, although it was
not until 1995 that the first condensate was
successfully produced in the lab.
This figure shows the velocity distribution
profile of the cloud of gas as it is cooled
down below 400 nano-Kelvin degrees. The peak
that starts emerging below this temperature
reveals the remarkable phenomenon known as
Bose-Einstein condensation, in which a large
proportion of the atoms suddenly occupy a
single quantum state, and this is precisely
where macroscopic quantum phenomena suddenly
become apparent. The colours indicate the
number of atoms at each velocity, with red
being the fewest and white being the most.
The areas appearing white and light blue are
at the lowest velocities.
Well, it turns out that the spread of velocities
– and hence momenta – in the peak is found
to be close to the minimum allowed by the
uncertainty principle. That is, spatially
confined atoms – where their position is
known with accuracy - have a minimum spread
of momenta distribution, below which the uncertainty
principle would be violated. On closer inspection,
the peak is observed to be anisotropic, with
a greater spread of momenta in one direction
than another. This can be explained by the
fact that the trapping region where the atoms
are confined is itself anisotropic. The direction
with the widest spread of momenta is that
with the narrowest spatial confinement, just
as one would expect from the uncertainty principle.
Thank you so much for watching, I hope you’ve
enjoyed watching this video as much as I’ve
enjoyed making it. There’s two or perhaps
three more videos coming in this series covering
yet more fascinating ideas around Heisenberg’s
Uncertainty Principle. I am currently working
on the scripts and the next video should be
ready in a few weeks. In the meantime, think
of this:
Quantum Mechanics, Lasers and Bose-Einstein
Condensates… How about using these three
ingredients to actually find out how to build
a proper lightsaber? Not this… but one these?
I am currently making a little extra video
for you, which will be available very soon
so stay tuned! Don’t forget to like, comment,
share and subscribe. And last but not least,
thank you so much to all my patrons and also
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and hopefully see you very soon!
