What exactly is the Heisenberg Uncertainty Principle?

Natural science, does not simply describe and explain nature, it is part of the interplay between nature and us – Werner Heisenberg

Some of you readers might have read about the Heisenberg Uncertainty Principle in high school or maybe even in college if you’ve attended courses on Quantum mechanics and Atomic Chemistry. But here we’re going to touch this topic both on an intuitive and mathematical level. Life has many trade-offs, the way my professor explained this to me is a very intuitive way of understanding this topic.

Let’s say you’re observing the routine of your mom and after a week, you realized that she’s gone to the Church on Saturday. Now, based on just this information, can we predict that your mom goes to the Church every Saturday? Of course not, we need further time for observation. Let’s say you then observe her for a month and notice that she goes to the church every Saturday. Based on this we might be able to make a statistical prediction, that your mom goes to the Church every Saturday. But still, we can’t be a 100% sure. Now let’s say you observe her for a year and see that she goes to the Church every Saturday. Based on these observations, we can say that there’s a high probability that your mom goes to the Church every Saturday, but we still cannot be a 100% sure as it’s entirely possible that your mom decides to go the school next Saturday. Hence, we see that we would require an infinite amount of time to perfectly predict whether your mom goes to the Church every Saturday. Thus, there is a trade-off between the frequency of her visits and time.

This is an intuitive example of a real-world situation, like what our beloved Heisenberg Uncertainty Principle says. Introduced in 1927 by the German physicist Werner Heisenberg, the principle states that the more precisely the position of a particle is determined, the less precisely it’s momentum can be predicted from initial conditions and vice versa. This can majorly only be applied in the quantum realm and cannot be applied to everyday macroscopic objects like you and me.

 So, basically what the Heisenberg Uncertainty Principle wants to tell us is that let’s say there’s an electron revolving around its orbit and if we accurately measure it’s velocity with 100% precision, we have absolutely no idea where the exact position the electron is located ! How cool is that! in such a case where we know a particle’s velocity, it’s position can be graphed as a probability function. Hence the exact location of the electron is not certain but is rather expressed as a probability. 

Mathematically we can say that the probability of finding an electron at a point within an atom is directly proportional to |Ψ²| at that point. |Ψ²| is known as probability density and is always positive. |Ψ²| can also be interpreted as the probability of finding an electron in a unit volume. Hence, the probability of finding the electron in an infinitesimally small volume dV = |Ψ²|dV. Ψ (pronounced as psi) is also called the amplitude function and has a few interesting properties that can be mathematically proven. It in continuous, single valued and finite. Also, since the probability of finding the electron in the entire volume that we’ve assumed is 1, we can also write the integral equation ʃ|Ψ²|dV=1 with limits from -∞ to ∞.

Ψ is basically the mathematical representation of waves in quantum mechanics and is called wave function, its value depends on the coordinates of the electron in the atom and doesn’t have any physical meaning. It is found in many other equations too including the Schrodinger’s Equation (that deserves a separate article on it altogether), radial probability density, radial probability distribution etc.

For some of you math freaks, here’s an interesting triple integral. Considering the position of the electron in three dimensions, we can also state that the probability of finding it in that three-dimensional space is 1. Hence, we have the triple integral, ʃʃʃ|Ψ²|dx dy dz =1 with limits from -∞ to +∞ in each integral. If a wave satisfies that triple integral, it is said to be a normalized wave function.

The concept of treating the electron both as a particle and a wave is called wave particle duality (this too deserves an independent article). Coming back to the Heisenberg Uncertainty Principle, we can mathematically represent it as Δx Δp ≥ h/4∏. Here Δx represents uncertainty in position and Δp represents uncertainty in momentum, h is called the Planck’s constant and its value is 6.62 X 10^-34 Joule second.

Therefore, this equation says that the product of uncertainty in position and uncertainty in momentum must always be greater than or equal to h/4∏. This essentially means that if uncertainty in position is very less i.e., we exactly know where the particle is located with a high degree of precision, then uncertainty in momentum must be very high to satisfy the equation i.e. we know very little about its momentum. And since momentum and velocity are closely related (p=mv), we don’t know its exact velocity in this case.

In the realm of quantum mechanics, we call variables such as position and momentum as conjugate or complementary variables. This in mathematics means that these variables such as position and momentum become Fourier transform duals. Other examples of conjugate variables would be time and frequency, Doppler and range etc. The important thing to be remembered is that the Heisenberg Uncertainty Principle can only be applied in the Quantum World on microscopic bodies showing wave particle duality. It cannot be applied on everyday macroscopic objects like you and me because macroscopic objects tend to have a high degree of particulate nature with an infinitely small wavelength.

Heisenberg formulated the Uncertainty Principle at Niels Bohr Institute in Copenhagen. According to Albert Einstein, randomness is a reflection of our ignorance and inability to comprehend a fundamental physical reality. On the other hand, Niels Bohr debated that probability distributions are irreducible and fundamental. Einstein and Bohr deliberated over the Heisenberg uncertainty principle for many years.

Heisenberg came to a very beautiful conclusion that formed as a base for further studies in the field of Quantum Mechanics. He theorized that absolute determinism was impossible as it required both position and momentum of the particle. This paved the way for the use of probabilistic determination of a particle due to the relation between position and momentum. This concept forms the base for the Copenhagen Interpretation of Quantum Mechanics.

A particularly interesting way to interpret the uncertainty principle is an illustration of how we see everyday objects. You can view and read this article due to photons of light which strike the surface and then bounce away. These photons inevitably transfer some energy to the surface it has bounced from. This energy will change the momentum of the electron on the surface. This way, we will be able to determine its momentum but not position. On the other hand,since quantum particles move at high speeds, it may no longer be in the same place. In either case, the uncertainly principle works and we cannot exactly determine the position and momentum of the electron.  

The uncertainty principle explains many phenomenons that cannot be explained using classical physics. For example, in atoms, negatively charged electrons orbit the positively charged nucleus. According to the theory of classical electrodynamics, the two opposite charges should attract each other eventually making the electron fall out of orbit and collapse in a spiral way. In such cases, the uncertainty principle makes things a lot simpler. If the electron gets too close to the nucleus, then its position can be precisely determined. But since its position can be accurately defined, its momentum would have a large error in measurement. Phew! Heisenberg saved us there. 

The uncertainty principle can also explain alpha decay. Alpha particle is a helium nucleus having two protons and two neutrons emitted by a heavy nuclei such as Uranium-238. These alpha particles are usually present inside the nucleus and require a very high amount of energy to break out. But since an alpha particle inside the nucleus has a well defined position, it doesn't have a well defined velocity. This means that there is a very small chance that it might escape from the nucleus. This result sin alpha decay and radioactivity is observed. This process is known as Quantum Tunneling.  

One of the most intriguing results of the uncertainty principle is about vacuums. Vacuum in general sense means the absence of matter. However, in the quantum realm, things are not that straight-forward. Here, Heisenberg's uncertainty principle is expressed in the form of energy and time. Thus, we can only accurately measure either the energy or time but not both. In this case, energy and time are conjugate variables. 

Uncertainty, which although seems very complex and some mystical theory, is fairly easy to understand and we probably wouldn't be here if this principle wouldn't exist. Werner Heisenberg received the Noble Prize for Physics in 1932 for his work in Quantum Mechanics. Another interesting thought example given by Albert Einstein disproves the Heisenberg Uncertainty Principle. Let’s say we have two balls, stick them together. Now spring them out so that both go in mutually opposite directions. Now, we can find the position of the first ball and not the momentum. Similarly, we can find the momentum of the second ball but not its position. But since the momentum of the second ball is mathematically related to the first ball, we can find the momentum of the first ball too! And voila, we hence can know both the exact position and momentum of both objects. I’ll leave this thought experiment open to interpretation to our readers. 

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