Since I was busy last week and I’m feeling ill this week, my good friend Michael Schmidt has agreed to write a guest post for me this week. Mike has a masters degree in physics from the University of Colorado, an interest in teaching, and a passion for math and physics. You can find out more about him on his personal website or read more on his blog, duality.io.

So, without further ado, here’s Mike’s article.

# Force Vs. Energy

When we teach physics, usually force is one of the first concepts. Force is easy to understand. I can have you imagine riding in a car riding around a curved road. As the car accelerates, the seat pushes you along. When the car turns you can feel the seat push you in the direction of the curve. In fact, force is such an understandable notion we often neglect to ask what force is or if there may be a better way to talk about the world.

## What is force?

Newton’s notion of force is the method which physically exchanges momentum. If two objects interact, they change each other’s momentum. Think of a two billiard balls bouncing off each other. If you placed your finger between between the balls you could feel a considerable force (don’t really do this, it will hurt). The billiard balls feel force due to the other and bounce off each other.

Now, this is how we speak about interactions for the most part. We draw force diagrams and use them to create equations we can solve. This, however, is not always so simple.

Let’s consider two similar examples where a ball bearing rolls (frictionlessly) down a slide: one where the slide is a straight slope and the second the slide is curved. Now suppose you want to find out how fast the ball will be moving when it gets to the bottom of the slide assuming it was nearly stopped at the top. In the first example at every point on the slide the effective force on the ball is constant. This is due to the slope being the same, what is true for one part is true for any other. Since the force is constant we can use the constant force equations to solve this.

Now, the second case. This situations is substantially more difficult, we need to recompute the force for every point along the slide.

We can’t use any convent equations we have to derive them. This can certainly be laborious and is not preferred.

Wondrously, there is a better way: **energy**.

## Energy

Energy is a strange notion; unlike force, you can’t feel energy.

The rules of Newtonian mechanics can be used to create two quantities: *kinetic energy* (or KE) and *potential energy* (or PE). Energy, unlike force which has a strength and direction, is just a number. *Kinetic energy*, roughly, is how much work it takes to accelerate an object up to some speed, whereas *potential energ*y is the capacity for an object to acquire kinetic energy. In other words, potential energy is energy that can *become* kinetic energy in the future.

Energy may flow between each of these types of energy but their total must always remain the same. To illustrate this imagine a spring fixed to a table on one end and let there be a weight on the other.

If you pull the spring to one side, stretching the spring, and release the weight it will move back and forth. When the weight is at the resting position of the spring, the weight will be under no force and will be traveling as fast as it can go, since as it continues to move it will be slowed again by the spring. It’s at this point the weight has it’s maximum kinetic energy and it’s minimum potential energy since the weight will not be sped up anymore. In contrast to this point, at both ends of the oscillation, the weight will stop. Here, we say the kinetic energy is zero and the potential energy is maximum.

If we use the notion of energy, we can make any situation like the bearing on the ramp nearly trivial to solve. This works since energy is allowed to be either kinetic or potential and the total must always be same. For the ball bearing on the slide example, the ball has only potential energy at the top of the slide and only kinetic energy at the bottom. We can represent this in an equation by

Since is an constant, you can make both sides equal for the beginning and end:

We can then solve for the and we would know the final speed of the ball. This method is has some obvious advantages, but all it seems we have done is find a quantity which hides the force.

## What is Energy?

Potential and kinetic energy seems just to be abstractions of force. In other words, energy isn’t real, the force is. We just made up energy to make the math easier.

This certainly seems like the right answer, especially in the light of how we can actually feel force and energy can only be referred to in equations. Of course, I would not have said that if it were so simple. Quantum mechanics seemed to turn the scientific world on it’s head but could the notion of force be false too?

# The Aharonov-Bohm Thought Experiment

This experiment begins with the double slit experiment, which shows the wave-particle duality of electrons. The double slit experiment has three elements to it: an electron emitter, a solid panel with two parallel cuts or slits in it, and a phosphorescent screen all arranged in this order. The setup is shown in the following image:

As the particles move away from the emitter they pass through the slits and interact to create the interference pattern show here:

The additional element added by the Aharonov-Bohm experiment is a very long solenoid encased within an impenetrable shell. The solenoid is place between the screens and the slits. A diagram for this is here:

This solenoid will create a magnetic field inside itself but not outside. This means under our view of things that there ought to be no changes to the setup outside the solenoid as the magnetic field cannot possible be exerting forces on the electrons. Interestingly, as you change the magnetic field strength the interference pattern on the screen will move. This effect is named the Aharonov-Bohm Effect after its discoverers. How could this be though, there is no force on the electrons! In fact there is no magnetic field anywhere the electrons are. The answer is there is another field present, the vector-potential. The vector-potential is a way to abstract the notion of a magnetic field and it is non-zero outside the solenoid. If it were just a mathematical trick, we would say it being non-zero outside is a side-effect of the math and is inconsequential. However, as we see the strength of this field has a direct impact on our observable world.

This questions our assumption that force is the most primitive or basic of interactions. Perhaps our mathematical trick is the real thing. There is much debate about this and there is likely no simple answer. The notion of force isn’t useless but it does have it’s limits. Maybe at some point a future experiment will help out understand more. For now, we’re stuck without an easy answer.

# Further Reading

If you would like to learn more about quantum mechanics Jonah has written a number of articles you might find interesting.

Jonah has a three-part series on quantum mechanics:

- In the first part, he introduces particle-wave duality.
- In the second part, he describes matter waves using the Bohr model of the atom.
- In the third part, he describes how one should interpret matter waves.

Jonah also wrote a post describing quantum tunneling.

More recently, Jonah wrote a two part series exploring the relationship between particles and waves.

- In the first part, he shows how you can build a particle up from a bunch of waves.
- In the second part, he shows that this isn’t always possible.