This episode is sponsored by Brilliant. Hi guys! Jade here. So a few weeks ago I
made a video on the Schrodinger equation and in it I said that if we place an
electron in a box the probability that it could be found outside the box is
zero, and a lot of you commented with questions like "oh but what about quantum
tunneling? Isn't there some cases where the electron can tunnel outside of the
box?", so I thought what the heck I'll just make a whole video about it. It's super
cool and I will answer that specific question at the end of the video, but
first, what is quantum tunneling? Well the short version is in regular classical
physics if you have a ball at the bottom of a hill, if it doesn't get a big enough
push to get over the hill it's kind of just stuck there. Putting this into
physics talk, if the ball doesn't have enough kinetic energy to get over the
potential energy of the hill, it'll never get over, like, ever. But of course, in
quantum mechanics things aren't so simple. If we replace the ball with a
quantum particle like an electron and the hill with some kind of potential
barrier, even if the electron doesn't have enough kinetic energy to jump the
potential barrier, sometimes it can end up on the other side. This is called
quantum tunneling and in this video we're going to see how it works. So now
the long version! So one of the biggest differences between quantum and
classical physics is that quantum physics is probabilistic. Unlike a ball
we can't pinpoint exactly where an electron is. This comes from the
Heisenberg Uncertainty Principle which says that we can never know the exact
position and momentum of an object. It's not because our measuring devices are
too crappy or because we're too slow, it's just something fundamental about
the laws of nature. But not all hope is lost! Maybe we don't know exactly where
the electron is but we know with a pretty high probability that it's around
here somewhere. We can actually model these probabilities with a wave or, more
technically, a wave function. This wavy cloud gives us the probabilities of
where the electron is likely to be, so now instead of imagining a particle
traveling toward a barrier, imagine a wave traveling toward a barrier.
Now when this wave collides with the barrier, because the electron doesn't
have enough kinetic energy to make it over, it gets reflected. But wait, what
about the whole tunneling thing? Well there's this secret property of waves
you probably didn't learn in school. Light is an electromagnetic wave so
let's imagine what happens when we shine a light beam through glass. When we shine
a light beam through a piece of glass, at the boundary where the glass meets the
air, the light beam will bend or refract. You may have noticed this effect if
you've ever looked at a straw in your water glass. The visual illusion comes
from the bending of light at the boundary of two different mediums, in
this case, air and water. But refraction isn't the only thing that can happen at
a boundary. Light can also get reflected. The amount of light which is reflected
and refracted depends on the angle that the light hits the boundary. All mediums
have a certain angle where 100% of the light beam is reflected. This is called
total internal reflection and you may have heard that when this happens 100%
of the incident beam goes back into the reflected beam, but that's not true.
These are Maxwell's equations and though they may look innocent they form the
entire foundation of classical electromagnetism. Remember how we said
that light is an electromagnetic wave? This means that the way light behaves in
different scenarios can be predicted and modeled by solving Maxwell's equations
now when we solve these equations for the case of total internal reflection we
get something very interesting this isn't that interesting instead of there
being an abrupt drop off where the light hits the boundary there's this very
quick exponential drop off this is shown by this term here I know this looks
super complicated and well it is so let's just get rid of all that
mumbo-jumbo here and just focus on the bit that matters this is a graph of e to
the power X which as you can see models exponential growth but the term in our
equation is e to the power negative which is simply the backwards version of
this exponential decay so we have this tiny little drop off wave here this is
called an evanescent wave which in my opinion is a very suitable name the word
evanescent means soon passing out of sight memory or existence quickly fading
or disappearing an evanescent wave is pretty much exactly what it sounds like
it decays incredibly quickly lasting only a few wavelengths before vanishing
so we can't usually see or detect it but if we place another material
sufficiently close to the boundary of the first sometimes the evanescent wave
doesn't decay completely to zero before hitting the next material so it can then
continue to travel onwards this is called frustrated total internal
reflection and I recommend looking up a demo on YouTube after this I would have
shown you in this video but for anyone who has read my Twitter bio you know
that experiments are not my forte I actually did try it and it just didn't
work my olds digression optics was one of my favorite subjects in university
and we did a lot of work on evanescent waves but I never really got a physical
intuition for why they're there the only answer I ever found is because Maxwell's
equations say so like when you solve the equations you end up with this decaying
exponential but other than that I can't really say a physical reason for why a
wave can't just abruptly stop at a boundary and change direction so if you
do please explain it to me in the comments ok digression over this wave
might be puny but it's the reason behind why quantum tunneling is possible
remember that we're trading our electron as a probability wave which means that
when it gets reflected here in evanescent wave forms at the boundary if
the barrier is thin enough sometimes some of the wave actually makes it
through so if some of the wave makes it through and this wave represents the
probability of the location of the electrons then there's some very small
but nonzero probability that our electron is over here even though this
probability is tiny because there are usually so many quantum particles
involved in any physical process the effects of Quan
tunneling a large enough to be essential to nuclear fusion in stars spontaneous
mutation in DNA and scanning tunneling microscopy it may seem stalling that
we're treating the wavefunction exactly like an electromagnetic wave it's hard
to imagine something so abstract like the probabilities of electron locations
as a real physical thing that travels and reflects and tunnels the truth is
scientists still don't know exactly what a wavefunction is they don't know
whether it's purely a mathematical tool we've created to help us predict things
about quantum objects or whether it's a real physical wave but what they do know
is that it can be modeled pretty much perfectly by wave mechanics when we
solve the Schrodinger wave equation for the electron inside the barrier we get
this exponential decay which is exactly what we would expect of an evanescent
wave speaking of Schrodinger's equation in my video about that I said if we
place an electron in a box the probability that it could be found
outside the box is zero and a lot of you are confused because what about
tunneling well the truth is I didn't specify this very well in a lot of
university degrees a particle in a box is the simplest case we use to analyze
Schrodinger's equation but it's actually a particle in an infinite potential well
so instead of this being a box look at it as a well with infinitely high and
thick walls in tunneling the barrier needs to be thin enough so that the
evanescent wave doesn't have time to completely decay to zero before reaching
the other side only then can it propagate onwards with infinitely high
and thick walls that's obviously impossible so yeah my bad didn't specify
infinity in my last video someone commented asking whether it was futile
to try and truly understand quantum mechanics without doing the math and my
immediate reaction was yes while you can learn the catchphrases and get an
overall gist of what's going on to really get that gut feeling of
understanding an intuition you need to work through the problems and see what
the equations tell you I didn't really understand the Schrodinger equation
until I solved it myself brilliant org is a learning website with
an entire course dedicated to quantum mechanics it starts with the very first
Berman's which reveals strange quantum behavior and takes you all the way to
shredding equation it has this interactive quiz style which I love
because you can work through problems at your own pace and check your
understanding at every step I actually just worked through these
quizzes on the mathematical foundations of quantum physics and had a few of my
own aha moments as some questions I'd had since University were finally
answered there are also tons of other courses specialising in math physics and
computer science brilliant is offering a 20% discount to the first 200 people to
sign up using this link just go to brilliant org slash up and Adam and
start learning quantum physics today thanks for watching guys I hope you
enjoyed the video it was actually the result of a poll I posted on the YouTube
community tab so if you would like to be included in those polls and vote on your
favorite topics then just click the notification bell this video is also
part of a quantum physics series I've got going on at the moment which I've
linked for you at the end of the video and in the description so until next
time bye