Harry Potter is one of my all time favorite movie series. Every fan knows that in order to get to
platform 9 3/4, at London’s King’s cross station, you have to get through a barrier.
What if I told you that this kind of magic is permissible in quantum mechanics? What if
I told you that when you were upset and banged your fist or head against a wall, there was a
non zero chance that your head or fist could go straight through that wall to the other side?
Now, the caveat is that for macro objects like your fist or body, to go through a wall,
the probability is extremely low. But,
this probability rises appreciably as you get to smaller and smaller objects,
so much so, that by the time you get to fundamental particles like electrons,
this tunneling effect can take place all the time.
In fact, quantum tunneling is responsible for life on earth because without it,
the sun would not shine. It also plays a critical role in our efforts to build
ever more powerful computers. And quantum tunneling could even be responsible for the very
existence the universe – It may have caused the Big Bang which occurred 13.8 billion years ago.
What is this mysterious quantum tunneling effect, where does it come from? And why
is it one of the most important phenomena in physics? The explanation is coming up right after this:
Before I get into the unbelievable explanation of quantum tunneling,
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Imagine that you have a tennis ball rolling back and forth in a frictionless box.
When the ball hits the side of the box, it just bounces back and forth.
The only way for the ball to get on the other side of the box is either to gain
enough energy to go over the wall, by perhaps you picking it up and putting it on the other side,
or by having enough energy to break through the wall. There is no other way
in classical physics.
What if instead of a tennis ball, this was a quantum object like an electron?
The rules of quantum mechanics are such that quantum objects have a wave-particle duality.
This means that what we think of as an electron particle actually behaves like a wave,
a probability wave. This means that its position is not a precise location in space. It is defined
by a wave function that can only tell us the probability of finding it a particular location.
And while you may think it is in the box in front of you, the wave function is
such that there is a non-zero probability that the electron can be outside the wall.
In fact, quantum mechanics shows that the wave function of a particle exists in all of space,
in the entire universe up to infinity.
So the astounding revelation of quantum mechanics is that
while you think this electron is in the box in front of you,…
…there is a non-zero probability that could be on Mars. The chance is very,
very, very small, but the important thing to know is that this probability is non-zero.
So, what governs this kind of behavior?
We can attribute this to the Heisenberg uncertainty principle. I would not say
it is a cause, but rather a description of its behavior.
Basically, this principle states that the uncertainty in a real particle’s position
times the uncertainty in its momentum has to be greater than a finite number. Practically
this means we cannot know with 100% certainty what the position of that electron is.
And the wave function of the electron, which gives us the probability of finding it at any location
can be found using the Schrodinger equation. So let’s take a closer look at that and figure out
how it shows us the probability of finding our electron outside the box.
The Schrödinger equation was developed by Erwin Schrödinger around 1926,
and it is the equation that describes the wave nature of matter. This is the time independent
form of the Schrodinger equation. The Greek letter psi in the equation is the wave function.
Although the wave function depends on both time and position, we’re going to look at
only how it varies with position to figure out how our electron will behave around a barrier.
The wave function can be both positive or negative, but the square is always positive.
And it so happens that the square of the wave function, which is a function of position x,
gives the probability of finding the particle at that position.
The first part is the kinetic energy operator acting on the wave function,
the second part is the potential energy operator acting on the wave function. After the equal sign,
it is the total energy acting on the wave function. The Schrödinger equation
is basically a statement of conservation of energy. It says that kinetic energy
plus potential energy equals the total energy—But instead of just energies,
we have the energy operators that are acting on the wave function of the particle.
Let’s now consider the case of a particle inside a box with a wall
that has a finite thickness and finite height.
We can solve for the wave function inside the box, but outside the barrier. And it
will look something like this, where l equals the following, m is the mass, E is the total energy,
C is the amplitude, and X is the location of the particle. This is a typical sinusoidal wave.
We can also solve the wave function for inside the barrier. Psi two in this case looks like this:
V naught is the potential energy. What the mathematics of the wave function inside the
barrier shows is that it decays exponentially. As x, the distance or thickness of the barrier
gets larger, the psi function becomes smaller, but it never reaches zero. This shows us that
the function is non zero in a finite wall. This means that our particle can end up on
the other side of the wall. And we can solve for the wave function outside the wall, psi three.
In fact, it will look identical to the wave function inside the wall. The amplitude
will just be much smaller. The amplitude indicates the probability,
so the lower amplitude just means the probability of the particle being there is lower.
Like I said earlier, the square of the psi function gives us the probability
of finding the particle there. So as long as the wave function is non-zero,
there is a non-zero probability of finding the electron there.
This is Quantum tunneling. It represents the nonzero probability of finding a particle
where classical energy conservation would prohibit it.
So Why doesn’t this math work for large objects like a tennis ball, or our bodies?
If you look at the wave functions, what you will find is that it depends on several factors,
the thickness of the barrier, the height of the barrier or V naught, because that
determines the potential energy required to overcome it, and the mass of the particle.
For macro size objects like a tennis ball or your fist, the mass is so large, and
the thicknesses of the barriers so great that the wave function will have an extremely small value,
making the probability extremely small.
In fact, if you do a rough calculation, for my hand or body to tunnel through the wall,
the probability will be about 10^-40, more 0's than atoms in my body, which is about 10^28.
This is statistically impossible, though not physically impossible.
So what are the chances that can you walk through the wall
and get to platform 9 3/4? – the answer is not zero.
The thinner the wall, and the shorter the wall is, and the lighter the particle is,
the higher the chance that it can overcome the energy required to tunnel through the barrier.
By the way, when I say barrier, it doesn’t necessarily mean a physical barrier, it can
be any energy barrier. This tunneling effect has huge implications in nature and technology.
For example, electronic devices are generally based on transistors.
A transistor can be made in different ways, but they all have the same general form.
There is a source on one side, a gate or barrier to control the flow of electrons to
the drain on the other side. If we look at a transistor from the side, we essentially
have the same situation as with the electron in a box. If the control gate becomes too small, the
electrons just tunnel straight though, ruining our circuit. In fact, modern transistors are so
small, that tunneling is an issue. This is why you cannot make transistor any smaller.
And this is a big roadblock for developing better more powerful, more miniaturized electronic devices.
In nature, quantum tunneling is responsible for the sun
shining. How you ask? Stars shine due to nuclear fusion.
Fusion is when you take two light elements like hydrogen and fuse them together
to make a heavier element, like helium. The process usually results in the release
of excess energy. Hydrogen bombs also work similarly.
The most abundant element in the universe is hydrogen,
and this is also what the sun mostly made of. It is its primary fuel source.
The sun fuses the nuclei of hydrogen atoms together to form helium. Larger
stars can produce atoms up to the size of iron using this same fusion process.
Tunneling is essential here because hydrogen nuclei are composed of positively
charged protons repel each other. In order for them to come together,
they have to overcome this extremely strong repulsion. This is called the coulomb potential.
In order for the sun to ignite, two hydrogen nuclei have to overcome this coulomb potential
in order for them to get close enough for the strong nuclear force to take over,
and fuse them together. This repulsion is an energy barrier.
But because of quantum tunneling, the probability for this process to occur,
becomes much more likely because the two nuclei can tunnel through this barrier. It’s like going
on the other side of a mountain through a tunnel instead of over it.
If it weren’t for quantum tunneling, most stars may never have ignited.
Although, the hydrogen atoms fuse into helium,
this helium can then also start fusing into bigger atoms in bigger stars,
where elements like carbon, oxygen, and nitrogen get formed. These are of course critical for life.
The most remarkable effect of quantum tunneling is the possibility that the universe itself
may have come to be because of it. The most accepted theory of the beginning
is that the universe came from an infinitesimally small point called a singularity, where all the
energy of the universe was initially located. It expanded exponentially through an inflationary
process that overcame any attractive forces such as gravity. This is better known as the Big Bang.
But the big mystery is what gave the universe the kick that it needed in
order to overcome the energy required to counteract attractive forces like gravity.
This is not well understood, and not proven, but according to calculations made by physicist
Alexander Vilenkin, the probability that the universe could overcome the energy barrier
needed for inflation to occur and the big bang to result, is not zero even at the
limit of the universe being size zero – that is, having no space and no time.
In other words, the big bang could come from nothing.
If this is true, then the process of quantum tunneling,
which comes directly from the most important concepts of quantum mechanics
is absolutely the most important natural process in the universe.
And I hope this video helped you understand this astonishing property of nature
a little bit better. I’d like to thank my generous supporters on Patreon. If you like
my videos, consider joining them.
I will see you in the next video my friend!