How can a photon which has no mass have momentum?

If you’re a fan of physics, or just someone who loves diving into the complexities of the universe, you’ve likely stumbled upon an intriguing question: How can a photon, which has no mass, have momentum? It’s a common question, and at first glance, it seems like a contradiction.

After all, momentum is defined as the product of an object’s mass and velocity. If photons are massless, how could they possibly have momentum?

In this post, I’ll take you through the deeper mechanics behind this question, and by the end, you’ll have a better understanding of not just photons, but how we think about energy and mass at the most fundamental levels.

We’ll cover classical physics, how it connects to modern theories, and ultimately, why mass is nothing more than a form of energy.

The Classical Momentum Equation

To begin with, let’s review the basics. In any introductory physics class, you’re taught the formula for momentum:

[ p = mv ]

Where:

• p is momentum,
• m is mass,
• v is velocity.

At this point, the problem is clear: photons have a velocity (the speed of light, ( c )), but their mass is zero. Using the classical equation, we would get:

[ p = 0 \times c = 0 ]

That seems to suggest photons have no momentum. Yet, physicists are confident that photons do carry momentum.

So how do we resolve this? Before getting to the answer, let’s examine the connection between momentum and energy.

Momentum, Energy, and Kinetic Energy

Kinetic energy is another concept closely tied to momentum. The classical equation for kinetic energy is:

[ KE = \frac{1}{2} mv^2 ]

And there’s a useful relationship between momentum and kinetic energy. By starting with the equation for momentum, we can derive something interesting. First, take the momentum equation:

[ p = mv ]

Now, square both sides:

[ p^2 = m^2 v^2 ]

Dividing both sides by mass ( m ), we get:

[ \frac{p^2}{m} = mv^2 ]

Does that look familiar? It’s essentially part of the kinetic energy equation. We can plug this into the kinetic energy formula, giving us:

[ KE = \frac{p^2}{2m} ]

In classical physics, this shows that momentum and kinetic energy are related. In other words, anything with kinetic energy must also have momentum. But photons still seem to be the exception, because they don’t have mass.

When Classical Physics Breaks Down

So far, we’ve only been talking about equations that were developed in the Newtonian era, where objects move relatively slowly compared to the speed of light. These equations, as useful as they are, don’t apply to objects moving at or near light speed, nor to objects that have no mass—like photons.

In fact, the classical equations for momentum and energy are special cases that break down when we’re talking about the fast-moving, massless world of light. Here’s where Einstein comes in.

Einstein’s General Equation for Energy and Momentum

When you think of Einstein, you probably think of his famous equation:

[ E = mc^2 ]

But guess what? This equation is only true for objects that have mass and are at rest. It’s not a general equation that works for everything, and it definitely doesn’t work for massless particles like photons.

The more general equation that describes the relationship between energy and momentum for all objects, whether they have mass or not, looks like this:

[ E^2 = p^2 c^2 + m^2 c^4 ]

Let’s break this down:

• ( E ) is the total energy,
• ( p ) is the momentum,
• ( c ) is the speed of light,
• ( m ) is the mass of the object.

In the case of photons, the mass ( m ) is zero. So, the equation simplifies to:

[ E = pc ]

This means that the energy of a photon is directly related to its momentum. Even though a photon has no mass, it still has energy, and that energy is linked to its momentum by the speed of light.

The Illusion of Mass

But wait—it gets even more mind-bending. You might think that mass is something special, something fundamental to the universe. Well, modern physics tells us that mass is an illusion.

Most of the mass of objects made of atoms (like you and me) doesn’t come from the particles themselves. Protons and neutrons, which make up the mass of atoms, are composed of even smaller particles. These particles move incredibly fast, and their mass is mostly a result of their energy in motion.

So when we say that an object has mass, what we’re really saying is that it has energy. And if a proton, which has mass, carries energy and momentum, a photon—being nothing but pure energy—can also have momentum.

What This Means for Photons

At the heart of it, a photon is nothing more than moving energy. Just like protons, photons carry momentum because they are energy in motion. The reason the classical equation ( p = mv ) doesn’t work for photons is because it was never meant to describe something that has no mass and moves at the speed of light.

The general equation that Einstein developed shows that energy and momentum are fundamentally linked, even when mass isn’t involved.

Why Does This Matter?

As you learn more about physics, you start to see that the rules we learn early on are just simplifications. The more you dig into advanced topics like relativity and quantum mechanics, the more you realize that the universe doesn’t work the way we intuitively think it does.

The fact that a photon, with no mass, can have momentum is just one small example of how mind-blowing the subatomic world really is. And, it serves as a reminder that every equation has its limitations. Physics is full of special cases, and understanding when an equation applies is as important as knowing the equation itself.

Final Thoughts

In conclusion, photons do carry momentum, even though they have no mass. This is because energy and momentum are deeply connected, and mass is just a form of energy.

The classical equations for momentum and energy are useful, but they only apply in limited situations. The true beauty of physics is how it continuously reveals deeper layers of reality as you explore more advanced concepts.

So, the next time you think about momentum, remember that it’s not just about mass—it’s about energy in motion. Physics is far more intricate than it seems at first, and that’s what makes it so fascinating.

If you enjoyed this deep dive into photon momentum and energy, make sure to share this post and continue learning. There’s always more to discover, and the more you learn, the more you’ll appreciate that physics really is everything.