# Natural Science #5: Uncertainty Principle

In the last natural science post, we briefly discussed the Heisenberg uncertainty principle. Considering that this principle laid the foundation for all of modern quantum mechanics, we could spend a little more time on it.

Simply put, the uncertainty principle states that there is a built-in limit to what one can know about a quantum system. For example, the more we know about a particle’s position, the less we can know about its momentum or speed, and vice versa. In the past, this has been confused with the observer effect, which states that the measurement of a system cannot be made without affecting the system. For example, the light photons that need to bounce off of an electron, for us to be able to see it, will actually change the momentum of the electron.

In reality, the uncertainty principle is a fundamental property of all wave-like systems and has nothing to do with the observer effect. If interested, an experiment showing this can be found here.

This concept is not very intuitive. For centuries, the idea that the state of the universe at one time determines the state at all other times was dogma. Newton based his laws of motion around it. The idea that there is no deterministic law in nature didn’t sit well with many, including Einstein who famously said:

God does not play dice.

But we are getting ahead of ourselves. Let’s try to break down the fundamental concepts behind the uncertainty principle.

### Math

Let’s get the math out of the way first:

Uncertainty Principle:

ΔxΔp ≥ h/4π

Δx = uncertainty in position
Δp = uncertainty in momentum (p=mv)
h = Planck’s constant (really really small number)

The product of the position and momentum uncertainties needs to be greater than or equal to a really really small number (6.63*10^-34). Planck’s constant, that really really small number, deserves a post of its own; but for now, we can just view it as the discrete measure of actions at the quantum scale.

To make things easier, lets say that the stuff on the right side of the equation equaled 1 and that our uncertainty for both position and momentum is 1. The inverse relationship between position and momentum uncertainties can be seen as follows:

(1)(1) ≥ 1
(0.5)(2) ≥ 1
(10)(0.1) ≥ 1

As we get more accurate with our measurement of one variable, the other variable becomes less accurate. This is true for all of reality, even at the macro level. For example, let’s apply this equation to a baseball that has a mass of 1kg, a velocity of 5m/s, and an uncertainty in momentum of .0001. Let’s solve for our uncertainty in position.

Δp = mv(.0001)
Δp = (1)(5)(.0001) = .0005

Δx ≥ (5.27*10^-35)/(.0005)
Δx ≥ 0.0000000000000000000000000000000105

The uncertainty principle doesn’t affect our day-to-day lives since being within 10^-35 is pretty accurate. This is all well and good, but it doesn’t really help us understand why the uncertainty principle does what it does. Let’s look at an experiment now.

### Single Slit Experiment

This video shows an experiment where photons are launched through a slit towards a target screen. At first, when the slit is largely open, the photons pass through in a straight line and hit the target as a dot. When the slit is closed more, it reduces the area that the photons can pass through. Another way of saying this is that the uncertainty in position decreased as the slit gets smaller. We know that when the uncertainty in position is decreased, the uncertainty in momentum will increase, and this is proven visually as the beam spreads out over the target screen.

We still don’t know why this is happening though. To answer that, we need to understand the matter-wave like make up of the world.

### Wave particle duality

The uncertainty principle exists because everything in the universe behaves like a particle and a wave at the same time. Particles exist in a single place at any instant in time. They can be thought of as a dot on a graph. Waves are disturbances spread out in space and do not have a single position. We describe waves by their wavelength which is related to momentum. For example, a fast-moving object has a high momentum and a short wavelength. In the same way, a heavy object has a high momentum and a short wavelength. The wavelike nature of everyday objects can not be seen because of these very short wavelengths. This is shown mathematically as follows:

λ = h/mv
λ = h/p

λ = wavelength
h = Plank’s constant
m = mass
v = velocity
p = momentum

At the quantum scale, these wavelengths are large enough to be measured. If we measure a photon as a wave, we know its wavelength and momentum, but we do not know its position. If we measure a photon as a particle, we know its position, but we do not know its wavelength or momentum.

Physicists get around this by combining waves of different wavelengths in order to get wave pulses at increasingly large intervals. For example, if two waves are combined, the areas where they align will get larger and the areas where they are opposite will cancel out. As more and more waves are combined, there is more and more flat area between the wavy regions. These wavy regions are called quantum object and have both particle and wave-like features. We can know both the position and the momentum, but at the cost of certainty for either.

The more waves that we combine into the quantum object, the more certain we will be about position but the less certain we will be about momentum. The less waves we combine into the quantum object, the more certain we will be about momentum but the less certain we will be about position.

This is visualized in the following video:

### Einstein and Bell’s Theorem

Einstein rejected the idea that there was uncertainty in nature. Instead, he put forth a thought experiment in which measuring an event at one corner of the universe would affect the outcome of measuring an event at the other corner of the universe. He called this thought experiment “spooky action” at a distance.

To understand this thought experiment, we need to understand particle spin. All particles have an angular momentum and an orientation in space. We measure this spin by choosing an orientation and then detecting whether the particle is aligned with (spin up) or opposite (spin down) the orientation. If the orientation we choose is perpendicular to the particle, then there will be a 50% chance of spin up or spin down and the particle will retain that spin.

The act of measuring the spin actually changes the spin of the particle. Einstein proposed that if two particles were formed spontaneously out of energy, we would know that one particle would have to be spin up and one would have to be spin down (conservation of angular momentum). If we measured one of these particles from a perpendicular position, where there was a 50% chance of spin up or spin down, the other particle would instantly be the opposite as a result of the measurement.

This is quantum entanglement, which just means that the spin of one particle is always the opposite of the other. If we measure one particle in any direction, we will instantly know the spin of the other, even if it is light-years away. This would imply that information was traveling faster than the speed of light. Einstein thought this was crazy.

He put forth a hidden variable theory that claimed there was an underlying reality, in which particles have well-defined positions and momentum. In this theory, the particles contained hidden information about which spin they would have under different measurement orientations. No signal would have to travel between them, instead they contained the information at birth.

John Bell proved that Einstein was wrong. Bell devised an experiment where entangled particles would be sent towards two detectors. The detectors orientation would be chosen at random between three states: 0 degrees, 120 degrees, and 240 degrees. Bell measured the amount of occurrences where the two detectors gave different results. For example, if one particle went through a detector with the 0 degree orientation and the other went through a detector with the 120 degree orientation, both particles could record spin up.

I won’t go into the math, but if the particles contained hidden information it can be proven that 55.6% of the time the detectors should show different results. If it was truly random, like the uncertainty principle claims, the detectors would show different results 50% of the time.

Experiment after experiment have gotten the 50% result. The uncertainty principle would seem to have been proved.

### Does God play Dice?

The results of Bell’s experiment are still debated among physicists. It could mean that all particles live in a state of uncertainty, and don’t have a spin until measured. It could also mean that faster than light information is being transferred between the particles through an unknown channel. As far as we can tell, that information is completely random.

Stephen Hawking noted:

It seems that even God is bound by the Uncertainty Principle, and can not know both the position, and the speed, of a particle. So God does play dice with the universe. All the evidence points to him being an inveterate gambler, who throws the dice on every possible occasion.

### As it relates to investing

Elroy Dimson, a professor at the London Business School, argued that:

Risk means more things can happen than will happen.

Howard Marks put it this way:

No ambiguity is evident when we view the past. Only the things that happened happened. But that definiteness doesn’t mean the process that creates outcomes is clear-cut and dependable. Many things could have happened in each case in the past, and the fact that only one did happen understates the variability that existed. What I mean to say is that the history that took place is only one version of what it could have been. If you accept this, then the relevance of history to the future is much more limited than may appear to be the case.

Both of these points are analogous to the uncertainty principle. We might think we know what will happen tomorrow, that the world follows a clear path, but in reality many possibilities are likely. People who get too confident in their predictions will always be blindsided when oil prices crash, when Brexit passes, when Lehman fails, or when any other possible future occurs.

### Conclusion

As hard as it is to conceptualize, reality is random. We can calculate probabilities, but we cannot make any definite predictions. The future is not governed by any single law. T.H. White, in his novel The Once and Future King, noted that:

Everything not forbidden is compulsory.

Unless something is forbidden, the uncertainty principle means it will eventually happen. This applies in both physical nature and in human markets.