Heisenberg Uncertainty Principle — Overview & Importance

$$\Large\color{#D73D75}{Uncertainty\ And\ Electrons}$$

Where, Oh Where Has The Electron Gone?

The start of quantum mechanics began when scientists tried to uncover the structure of atoms. Inside atoms there exists electrons and scientists strove to prove the location of electrons within atoms. Then in 1927, Werner Heisenberg demonstrated that the exact location of an electron and the path it takes is impossible to determine. This idea became known as the Heisenberg uncertainty principle.

Explaining The Uncertainty Of The Location Of Electrons

Locating a single electron at a specific point in time is in fact impossible. Let's imagine we try to locate an electron. One way would be to use a photon of light to interreact with the electron and bounce back, revealing the location. This would be like how bats or dolphins use echolocation. However, when the light interacts with the electron, energy would be transferred to the electron. This energy transfer would cause the electron to move faster. While trying to determine the location of the electron, we are changing the position. Therefore it is not possible to determine an electrons exact location.

Mathematics Of The Uncertainty Principle

Expressed mathematically, the Heisenberg uncertainty principle is:

$$(\Delta x)(\Delta m \nu) \ge \frac{h}{4\pi}$$

where:

• $\Delta x$ is the uncertainty of the electron's location

• $\Delta m \nu$ is the uncertainty of the momentum

• $h$ is Planck's constant equal to $6.626 \times 10^{-34} J\cdot s$

This equation reveals that we cannot know both the location and momentum of an electron. If we know the momentum with more certainty (smaller $\Delta m \nu$), then the location is less known (larger $\Delta x$). The reverse is also true. This applies to very small objects and particles on the atomic scale.

Heisenberg Uncertainty Principle: What does it tell us?

(Video) What is the Heisenberg Uncertainty Principle? - Chad Orzel

By TED-Ed

In this video, Dr. Orzel breaks down the Heisenberg Uncertainty principle by identifying the difference between particles and waves. He argues that it is easy to find the location of a particle, but it lacks momentum. On the other hand, he points out that it is easy to measure the momentum of a wave, but it is hard to assign a single location. He then explains how to measure both position and momentum of a particle. He concludes that this principle is not a metric that determines how precise the measurement is; rather, it is a result of the wave particle-duality.

We're exploring the origins of quantum mechanics. So far, we've studied how scientists began to shift away from classical physics. It started when Max Planck investigated blackbody radiation. Planck's quantum theory taught us that atomic energy levels are quantized. Quantization helped Einstein solve the photoelectric effect. Suddenly, we were uncertain if light was a wave or a particle. So, we developed the wave-particle duality. Around that time, Neils Bohr applied quantization to the atom. Bohr's atomic model was the first step toward theoretically predicting atomic spectra and absorption and emission. But, the quantum mechanical model would prove superior. Next, Loius de Broglie's equation linked the wave-particle duality to all matter. He connected Einstein's famous relativity equation and Planck's quantization equation. So, we realized that everything with mass has a wavelength.

All that we've studied so far was the lead-up to quantum mechanics. 1925 and 1926 would redefine physics. Two famous scientists devised the mathematical foundations for quantum physics. The first was Warner Heisenberg. He developed matrix mechanics, which describes quantum systems using matrices. A year later, Erwin Schrodinger conceived of wave mechanics. That's when he introduced his famous Schrodinger Equation. The models are equal but with different mathematical operations.

Today, we'll look at Heisenberg's most outstanding contribution. In future lessons, we'll examine Schrodinger's contributions.

The Uncertainty Principle

As Heisenberg was developing matrix mechanics, he discovered a unique phenomenon. Because electrons are quantum particles, they do not behave classically. As an example, let's consider an electron's motion. Electrons do not travel around the nucleus via circular motion. That was an approximation from the Bohr model orbits of the hydrogen atom. It worked because it was a one-proton, one-electron system. So, the electrostatic interactions acted classically. But it was a limited model. Instead, an electron's motion spreads throughout the space around the nucleus. In Schrodinger's model, we would call it an atomic orbital. But, the spread-out movements mean we cannot sharply define its position. So, there is an associated uncertainty.

The uncertainty principle is an inequality. It's written as $(\Delta x)(\Delta p) \geq \frac{h}{4\pi}$. We keep seeing Planck's constant ($h$)! That's because, at the quantum scale, everything gets quantized. So what does the inequality tell us? The product of the uncertainties in position ($x$) and momentum ($p$) is always greater than or equal to a constant ($\frac{h}{4\pi}$). That means, at the quantum scale, we can never have zero uncertainty. It also means that the uncertainty of position and momentum are proportional. If we are very certain of the position, we are more uncertain about the momentum. The reverse is also true.

At this point, you might wonder how we got the uncertainty principle. But that requires understanding some math.

The Basics of the Math Underlying the Uncertainty Principle

To understand the uncertainty principle, we need its mathematical foundation. Thankfully, the concept is relatively simple. To actually perform the math is unbelievably complicated. So, in this explanation, we'll only explain the big picture.

First, remember that quantum mechanics is about probability. A wavefunction is a mathematical function that describes a quantum system. So, our position wavefunction tells us the likelihood of finding our particle in a specific location. On the x-axis, we have the position. On the y-axis, we have the probability. The graph will peak at the spot we would classically measure. That's the most likely location. But, at the atomic scale, we can't be exact. How do we get the uncertainty? It's the width of the probability function.

It turns out that some variables are physically related to each other. The fancy term is conjugate variables. The uncertainty principle can actually link any pair of conjugate variables. The most common set is position and momentum. But another possibility is energy and time.

So, how are conjugate variables related? There is a mathematical process called a Fourier Transform. The Fourier Transform of position is momentum and vice versa. The math is quite complicated. But the concept is simple. Any mathematical function can get broken down into smaller component functions. The sum of the smaller components equals the original. But, there is a trick. The component functions are sine waves with different frequencies. The addition of the sine waves is an application of constructive and destructive interference. When the peaks and troughs align, they add. When they offset, they subtract.

So, what is the transformation? You plot the number of each component function versus the frequency. For example, let's say we had a position wavefunction made of one $sin(x)$ and three $sin(2x)$ functions. How would we plot the transformation? The frequency of the $sin(x)$ is $\frac{1}{2\pi}$. So, on the x-axis, we would go to $\frac{1}{2\pi}$. There is only one. So, on the y-axis, we plot one. The frequency of our other function is $\frac{1}{4\pi}$. We had three of that component function. So, we would plot $x=\frac{1}{2\pi}$ and $y=3$. But what have we done? We created a new function. In this case, we transformed the position function into the momentum function.

Wait, what about the uncertainty principle? There is a unique property of Fourier transforms that generates it. If the original function is broad, the transform will be narrow. The opposite is also true. If the original is thin, the transform is expansive. The most basic example is a sine wave, $sin(x)$. A sine wave is already a fundamental unit. So, a sine wave can only get constructed via one component sine wave. But we know a sign wave extends across all space with a frequency of $\frac{1}{2\pi}$. Why? Because $x$ can have any value. But, if we took the Fourier Transform, we would only have one spike. So, the original function is infinitely broad, and its transform is infinitely narrow.

Have you made the connection to the uncertainty principle? Our uncertainty is the width of our function. So, a very wide position wavefunction has considerable uncertainty. But, the momentum wavefunction will be narrow. So, it has a small uncertainty. The opposite is also true. So, the more certain we are about one variable, the less confident we are about the other.

Heisenberg's Physical Description

Heisenberg developed a qualitative description to help students understand his uncertainty principle. Imagine we want to measure the position and momentum of an electron. Electrons are unbelievably tiny. So, we don't have an easy way to measure them. What can we do? We can hit them with light and record the diffraction. But there is a problem! Because of the wave-particle duality, light also has position and momentum. So, its collision with the electron influences the electron.

Let's say we use light with a very narrow wavelength. We would have a small uncertainty in the position. But, we know that results in significant broad momentum wavefunction. In other words, our momentum is highly uncertain. In this case, the collision transfers momentum to the electron. So, we can't determine both simultaneously. The opposite occurs if we use a long wavelength. The momentum transfer is minimal. But, we get very little information about the position.

To be clear, please understand that Heisenberg's description is a teaching tool. It's one way to illustrate why the uncertainty principle exists. It is not the uncertainty principle in itself. The uncertainty principle is a fundamental aspect of quantum mechanics. The mathematics of conjugate variables and wavefunctions creates it. Many people have applied the uncertainty principle and this illustration to metaphysics or philosophy. That's fine. But, its different then the scientific method and often confuses introductory students. So, please be careful.