If I asked you to name a behavior or type of motion that is the most important motion in
the universe, what would you choose? Linear motion, like when you drive your car?
How about rotational motion, like that of the planets orbiting the sun? Sure that’s
pretty important. But I’m going to contend that the most important motion is that of
a spring. That’s right, like the spring in your mattress, or in the suspension
of your car. Why do I make such a claim? Well, classical springs like you might
have in your mattress have what’s called a harmonic oscillation. And this behavior has
a quantum mechanical analog in what’s called a quantum harmonic oscillator.
And this kind of motion can model the motion of all fundamental particles
in the universe, as well as the forces mediating the interactions between them.
In fact, the prevailing fundamental theory of all matter and forces, called Quantum Field Theory,
is built upon this type of motion. What am I talking about? And how is a simple
spring related to our understanding of the entire universe? That’s coming up right now!
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Since I’m claiming that everything is a spring, to start, let’s look at the motion of a spring.
Imagine a wall. Let’s attach a spring to it horizontally. At the end of the spring,
we place a ball. We’re going to ignore gravity and friction in this example.
Right now nothing is happening. But if we give the ball a tap, it’ll start to move. The spring
starts oscillating. It goes back and forth. In the real world there will be some dampening
effect due to friction of the air and metal that will eventually cause it to stop oscillating.
But if we ignore that for the moment, then after tapping, it will keep oscillating forever.
This is called a simple harmonic oscillator. The movement goes from left to right crossing
the center where it was at equilibrium, the point at which the ball was before tapping it.
If we stop the spring, it’ll be at the equilibrium again and not moving. If we
think in terms of energy, it would be at its lowest energy and not moving.
What if we pull the ball distance x1 from the center and let go. Then
it will oscillate to the left and right up to the distance x1 from the center.
Note that where the potential energy of the ball is highest, it is not moving, so its
kinetic energy is zero. But where it’s moving the fastest, at the center, its kinetic energy is the
highest. The total energy of this system is the sum of the potential energy and kinetic energy.
We can write an equation for total energy, E, like this:
The first term is the kinetic energy, and the second term is the potential energy. P is the
momentum and m is the mass. Momentum is mass times velocity. k is a constant,
called the spring constant and x is the distance of the ball from the center.
Notice that while the potential energy and kinetic energy terms vary, with no loss to friction, the
total energy remains the same as the ball moves. This might not seem like a big deal so far,
but what’s important to understand is that this spring behavior is everywhere!
Think of a pendulum, it also goes from side to side with the exact same type of motion!
Same for a Kids swing on the playground, the same simple harmonic motion. The higher they swing,
the faster they will move at the center close to the ground.
But there are some other harmonic motions that might not be so obvious. If you’ve
ever played a stringed instrument, like a guitar, the string is oscillating just like
the spring. It’s also a harmonic motion. So these are interesting examples, but I
said everything is a spring! What do I mean? Well so far, we’ve looked at only classical
examples. But what happens if we now go to the quantum realm?
Well, the equation of energy for a quantum particle looks almost identical. Except,
now it is expressed as, H, the Hamiltonian, instead of E, energy. The difference is that
the Hamiltonian is an operator. It has to operate on something to
determine how its energy is evolving. In the case of quantum particles, the Hamiltonian operates on
the wavefunction, since the wavefunction shows how a particle evolves over time.
This is the Schrodinger equation, which is an equation describing the energy of
a quantum system. The Hamiltonian operates on the wavefunction psi
which then determines its total energy E. What we can do is replace k with omega, the
angular frequency using the following relation: The Hamiltonian, then becomes the following:
And when we plug this back in the Schrodinger equation, it takes the following form:
We have a psi on both side of the equation. To reinterate, psi is the wavefunction and
describes how the state of a quantum system evolves over time. And the square of this
wavefunction give us the probability of finding the particle in a particular region of space.
But Arvin, did you know Schrodinger’s equation isn’t actually a wave equation?
Sure, its solutions are called “wave functions,” but that doesn’t make it a wave equation.
If you look closely, you’ll see their forms don’t match.
The types of changes aren’t the same.
Wave equations and wave functions have the word “wave” in them for entirely
different reasons. Yes, I’m serious.
Schrodinger’s equation is actually a heat equation.
The motion it describes is flow, not waves.
If you want to learn more about that, come check out my video on the Science Asylum.
After you’re done with Arvin’s video, of course.
Nick is right, the Schrodinger equation in not a wave equation. A wave equation
encodes a wave. The wave function encodes the wave-particle duality of a quantum object. This
distinction might be a bit confusing, so for clarity check out Nick’s video
in the Science Asylum channel right up here: This equation then represents how
the energy of a quantum system evolves in time. What we have achieved here is the quantum analog
of classical springs. But in the quantum version, the properties of the spring,
like its momentum and position are probabilities determined by PSI, and not specific numbers.
But this format for spring-like behavior expressed as the Schrodinger equation is not the end of the
story, because the Schrodinger equation has limitations. The more complete description of
the universe is quantum field theory or QFT. QFT describes all particles as excitations in
quantized fields that are constantly in flux. And unlike the Schrodinger equation, it accounts for
things like particle creation and annihilation, radioactive decays, and particle interactions.
So how do we show that the same spring-like behavior applies to quantum fields?
We can do this by first rewriting the equation for the quantum spring in terms of energy instead
of position. To do this, we introduce the terms a and a dagger which represent the increase and
decrease of the energy of the spring respectively. Keep in mind that since this is a quantum system,
the energy can only increase or decrease by one quanta at a time.
These a and a dagger terms are related to omega, and x by the following equations:
By substituting and doing the math, We can rewrite the Hamiltonian of the quantum harmonic
oscillator now using a and a dagger like this: This equation is exactly the same as what we
started it, just rewritten in terms of increasing and decreasing energy. Btw,
this equation also shows explicitly that the energy can never be zero because of the constant.
Now here is the astonishing part, as it turns out, if you write a quantum field theory for a simple
field, then you get the following Hamiltonian. So this is almost exactly what we have in
the quantum harmonic oscillator or quantum spring. But now it’s for a particle field.
What’s different is that in the case of quantum fields, since they extend everywhere
in the universe, it has to be integrated over the three dimensions of the universe.
That’s what the d-3 term represents – we are integrating the momentum over three dimensions.
We are now not literally describing springs, but we are modeling particles in quantum fields which
turn out to have spring-like behavior. Now a and a dagger can be interpreted as operators that create
and annihilate particles in our quantum field. In addition, unlike in quantum mechanics,
where the number of particles are fixed, in quantum field theory, the number of particles
in the system can change. We can add particles by adding energy to the field, or annihilate a
particle by removing energy from the field. This aspect of quantum field theory is
incorporated in the “a” and “a dagger” part of the equation. They are also known as the
creation and annihilation operators. “a” is the creation operator and “a
dagger” is the annihilation operator. The math is not trivial and it’s easy
to get lost, so let me give you an intuitive understanding of it.
What this equation shows is that quantum field theory is essentially an infinite collection
of harmonic oscillators or springs, extending out to all of spacetime in all 3 dimensions.
We have our quantum spring. Because it’s quantum it will never be still, so there
is always some small oscillation or vibration in the ground state, which is the lowest energy state.
This minimal oscillation represents quantum fluctuations, also known
as virtual particles that come in and out of existence. You can think of this
as the spring never being fully at rest. Now if we give the spring a punch on the
order of Planck’s constant, it will excite the spring to the next state, the one particle state representing one particle.
Unlike the virtual particle state, this higher energy state creates a real measurable particle.
The spring now has a higher amplitude to indicate the extra energy we have added.
We can punch it an infinite number of times to increase the state to equal to 2, 3,
4 and so on particles. The amplitude can only be integer multiples of the Planck’s constant.
This is the quantum world; everything is in steps ensuring, we have a discrete number of particles.
We can never for example, have half the energy of Planck’s constant, representing half a particle.
Now we can add more springs in a line and connect them with rubber bands. In
this case they will now be coupled to each other, this a feature of a quantum field,
representing interactions and movement. We can for example punch one of our
springs from one side, and this energy will then propagate across the springs
like a particle moving in spacetime. We can now imagine that this mattress is
of infinite size. We can have waves representing particle motion and some of the waves can collide
illustrating particle interactions. And if you can imagine 12 mattresses
of infinite size, these would represent all the matter particles of the standard model.
Furthermore, we would have more mattresses that would act as intermediaries to exchange
energy from one mattress to another. These represent the bosons of the standard model.
So in quantum field theory, everything is essentially just springs.
But unlike the classical springs, the energy levels of the quantum field oscillators are
not continuous. They are “quantized”, that is, their oscillations are in discrete quantities
proportional to Planck’s constant. These discrete units, when localized,
manifest themselves as particles. You might ask what does this all
mean? Does it mean that all the particles in the universe are literally springs?
No, just because the math is that of a spring does not mean they are literally springs.
So if a spring is oscillating between its maximum potential and kinetic energy, what can we say is
oscillating in quantum field theory. I think what we can say is there is an oscillation in the state
of the particle, that is, there is an uncertainty built into quantum mechanics. Properties like
location and momentum is uncertain. So one way to think about it is that the
oscillation represents the quantum state of the particle itself. It’s not fixed. It is uncertain.
It's in flux. It's important to keep in mind that
the spring analogy is an approximation. It helps us solve some of the unsolvable math
of QFT by approximating certain motions as that of a spring. And that can help us get results often are
good enough for most applications. And that's the best we can ask of our math.
I’d like to thank my Patreon subscribers,
who help pay for these animations. I really appreciate it. I will see you in the next video my friend.