# Development of Quantum Physics VI – Heisenberg Challenged Our Worldview …Again!

Let’s have a quick flick through our story about the development of quantum physics and highlight some of the so far discussed key aspects!
The very beginning of quantum physics is represented by Planck’s quantum hypothesis which allows us to understand the black body radiation correctly. Einstein made use of this hypothesis by explaining the photoelectric effect. Later, de Broglie postulated that we could apply the wave theory to traditional “particles” like electrons etc. The previous part of this series stated Born’s suggestion to interpret the matter wave of a “particle” as a probability density function which embodies just the particle’s probability to be found at a certain place at a given time.
Today we will focus on a logical consequence of the wave description of matter. The resulting principle has become quite famous (at least it’s name): Heisenberg’s Uncertainty Principle.

(You can find the other parts of this series of articles here, if you haven’t read them yet.)

* * *

The uncertainty principle which was formulated by physicist Werner Heisenberg in 1927 is nothing more than a logical consequence of the wave description of matter (which consists of particles in the classical sense).

In order to develop our train of thought, let’s start with a wave packet. (A brief retrospection: In the context of quantum mechanics, a particle is mathematically described by a so-called wave packet. This is described in the fifth part of the series.) If we define the wave packet’s width – let’s call it $\Delta x$ – at the very point at which the function dropped to the $1/\sqrt{e}$-fold value and if we label the width of the amplitude distribution between the different wave numbers $k$ as $\Delta k$, then we get (as the result of some calculations) a relation as follows:

$\Delta x\cdot \Delta k=1$

In words:
The product of the spatial width ($\Delta x$) of the wave packet and the width of the wave numbers’ intervals ($\Delta k$) of the wave packet-forming matter waves is equal to 1.

Interestingly enough, this relation was already known from (classical) optics – it isn’t new in this sense.
The parallels between the new, wave-like description of “particles” and classical optics are of course no accident! It’s actually quite obvious that we can find aspects of one theory in the other one and vice versa – just because of the simple reason that we describe particles by waves and that light is a multitude of electromagnetic oscillations (= waves). It’s very convenient (and also cool, at least for physicists) that we can use the same mathematics for both descriptions!

The impact of the equation $\Delta x\cdot\Delta k=1$ on quantum mechanics becomes clear when we use the de Broglie relation for the particle’s momentum ($p=\hslash k$) and substitute for $k$: We obtain $\Delta x\cdot\Delta p=\hslash$. Heisenberg was able to show that this applies to all kinds of particles. Furthermore, he found out that the product of the spatial uncertainty $\Delta x$ and the uncertainty in momentum $\Delta p$ is always greater or equal $\hslash$. ($\hslash$ is Planck’s constant divided by $2\pi$.)

The famous version of the Heisenberg Uncertainty Principle therefore reads

$\Delta x\cdot\Delta p\ge\hslash$

Friedrich Hund, Werner Heisenberg, and Max Born in Göttingen, 1966
(Credit: Gerhard Hund)

Simply put, this inequality says that it’s impossible to simultaneously measure the position and the momentum of a particle with arbitrary accuracy – even with the best measurement devices possible. Nature dictates fundamental limits!

However, $\hslash$ is an unbelievably tiny natural constant (about 10-34 J·s). Its tininess is the reason why we usually don’t notice Heisenberg’s Uncertainty Principle in our day-to-day lifes. It seems that we are not subject to limits in simultaneously determining a particle’s position and momentum, but all this is just an illusion due to $\hslash$ being so dwarfish.

De facto, the Heisenberg Uncertainty Principle is true, although we can’t see, feel, hear, smell, and taste it most of the time.
Derek Muller from Veritasium deals with this subject in one of his videos (which, by the way, are all fantastic and really worth watching!) and even makes the Uncertainty Principle visible!

So, enjoy this concluding video: Heisenberg’s Uncertainty Principle Explained.
(Derek uses another version of the Uncertainty Principle: $\Delta x\cdot\Delta p\ge \frac{h}{4\pi}$. In doing so, he just makes use of a different definition of the wave packet’s width. But you won’t have any troubles in understanding the basic idea, which is that a lower limit in determining position and momentum of a particle exists!)

__________

 Previous part: We Cannot be Sure About the World, Physics Says Next part: Atoms Are Choosy When It Comes to Wavelengths