WHAT IS HEISENBERG UNCERTAINTY PRINCIPLE?
Nowadays, more and more people trust science and his achievements, take for example this last year,
shocked by the diffusion of Covid 19: the entire world is orienting its decisions
according to what scientists are discovering. If we skip to the past the situation was very
different, science wasn’t so developed like today and people used to not believe
in everything experts were saying, think for example to the Greek belief in “fate”.
The transition between these two opposite views on life has happened in the 18th and 19th century:
the period of Illuminism and Determinism. Although this revolution of human thoughts
and lifestyles has certainly brought to a great technological progress that continues until now,
many conclusions made in that era are totally wrong. Trusting the scientifical method is a
right thing to do, but posing all our hopes in science and thinking that everything can be
pre-determined through mathematic equations is an inaccurate reasoning that comes from Illuminism.
As a matter of fact, some great discoveries made at the beginning of last century unveiled a secret
and totally counterintuitive behavior of nature. I’m talking about quantum mechanics,
particularly about Heisenberg uncertainty principle: a shocking result that has changed
completely our way of thinking about practically everything in this world.
The whole 20th century was “affected” by this discovery,
which totally destroyed our previous vision of science which was characterized by determinism.
Hundreds of poets and writers have been influenced by the revolution of quantum mechanics.
But what is it about? Who was Heisenberg? What does his principle state? And why did it
have so many consequences even on different fields such as literature and philosophy?
These questions are the topic of this video, so if you are interested stick with me and prepare to
this incredible journey in quantum mechanics. (roll intro)
Heisenberg principle is amongst the most pivotal ones in physics
and it hasn’t to be violated in any possible experiment. In few words, it states that it’s
not possible to measure both position and momentum of a particle with infinite precision: the product
of the errors in the measurement of these two quantities must be higher than a constant value.
The result is that if you know where a particle is, you ignore its velocity and viceversa.
Unfortunately, as you can imagine, it’s not so easy to understand why this principle works,
so we’ll firstly try to look at what physicists thought before the development of
quantum mechanics. Given these bases, we’ll then go through the uncertainty principle
and we’ll try to understand why it’s so in contrast with previous theories.
What was physics like before the 20th century?
We’ll many things can be said, but to start our description we need to start from Newton.
Known as one of the three/four most brilliant scientists ever, Sir Isaac developed what is
recognized as “Classical Mechanics”, together with other famous studies on optics and obviously about
gravity. What are the main principles of this theory? The first one is the law of inertia:
“if a body is at rest or moving at a constant speed in a straight line, it will remain at rest
or keep moving in a straight line at constant speed unless it is acted upon by a force”. The
law of inertia was first formulated by Galileo Galilei for horizontal motion on Earth and was
later generalized by René Descartes. The second one is probably the most famous equation of
physics and it states that the force acting on an object is proportional to its acceleration
through the mass of the object itself: F=ma. The third is not relevant in this case
but it’s always necessary to be remembered: If an object A exerts a force on object B,
then object B must exert a force of equal magnitude and opposite direction back on object A.
Another major achievement made by Newton and his subsequent colleagues was the development of
mathematical calculus: derivatives and integrals are strictly linked to classical mechanics.
Even if we’ll not go deep in this subject, it’s necessary to underline the fact that mathematical
tools can be used to find the velocity and the position of a moving body, starting from its
acceleration. What is the consequence of all I’ve said? If we are able to calculate the mass of an
object and the force exerted on it, it’s possible to derive the acceleration through Newton’s second
law and then Maths will “provide us” its velocity and its position depending on time:
this is totally a deterministic model. As a matter of fact, Illuminism was powered
by the discoveries made by Newton about physics. However, physicists of that period knew that many
phenomena were not explicable by this classical theory, but when major discoveries were made in
the 19th century regarding electricity and magnetism the idea that everything could
be determined by science drastically raised. In fact, physicists like Maxwell and Faraday
understood that electricity and magnetism were two sides of the same coin, they elaborated the
electromagnetic theory that stated the existence of electromagnetic waves. Furthermore, all the
optics studied in the previous centuries was just a consequence of the four Maxwell equations that
governed electromagnetism, because light was just a particular type of electromagnetic wave.
What was the consequence of all this? Almost every phenomenon on Earth can be totally
explained with gravity, classical mechanics or electromagnetism: it was the apex of determinism.
Therefore, in 20th century physicists understood that to explain the behavior
of atoms it was necessary to completely change the way of thinking about the world, it was necessary
to abandon determinism. Classical theories weren’t able to explain why we exist, since they predicted
that medium life of an atom was just around some fraction of a second: totally incorrect!
To better describe a particular phenomenon called photoelectric effect in 1904
Einstein stated that light is quantized in small packets of energy called photons,
in some cases waves can be treated as particles with discrete values of energy. More precisely,
each photon has an energy proportional to its time frequency through a constant called Planck’s
constant. After that, in 1924 the famous Louis De Broglie tried to invert this correlation:
he proposed that all particles could be considered also as waves with a wavelength equal to Planck’s
constant divided per its momentum (which is the product of mass and velocity) ----- λ=h/mv.
So, the higher the momentum of the particle (heavy or fast objects) the shorter the wavelength.
And talking about waves we must mention Fourier transforms.
Who invented this mathematical powerful tool? It was elaborated by a French mathematician called
Jean Baptiste Joseph Fourier. After having fought in the French Revolution
he attended the Ecole in Paris and followed the lectures of great teachers like
Pierre Simon Laplace and Joseph Louis Lagrange. But wait a moment, what is a Fourier transform?
Don’t worry, we won’t deep in the mathematical tools used in this topic,
but we’ll just try to have an intuitive idea of what it is.
Fourier transform of a light wave is the distribution of the components of its time
frequency. The principle is that all functions can be reproduced by a continuous infinite sum
of waves with different frequencies (that, for experts, is an integral). Each frequency component
has a weight that is established by the Fourier transform. For example, the transformed of a solar
ray will have a peak on orange frequencies and decreasing values for other colours.
Similarly to this, the Fourier transform of a particle wave will be a distribution of spatial
frequency, which is proportional to momentum thanks to De Broglie’s Hypothesis. So, the Fourier
transform is able to change our perspective regarding the particle from position to momentum.
Even if we won’t deal with the mathematical expression of this topic,
we must underline its most important properties. If the function of which we calculate the
transform is defined on every possible position, its Fourier transform will be
limited to a discrete collection of frequencies. On the other hand, if the function is limited in
a region and zero elsewhere, its transformed will be continuous and the function will be
composed of an infinite number of frequencies. Think for example of an ideal yellow light ray
that travels from minus infinite to plus infinite: he will be composed just of a spike on one single
frequency correspondent to the yellow colour (with a subsequent discrete Fourier transform).
But what does it happen if our light ray is only existing in a region and zero elsewhere?
Well, the transformed will always have a peak on the yellow frequency,
but it must be composed of other waves that can zero the amplitude in all the other regions of
space (so the transformed would be continuous). You can naturally reuse the same considerations
with a wave particle like the ones theorized by De Broglie: the smaller the spatial distribution
of the function we want to transform, the broader the distribution of momentum will be.
Are you following these concepts? We’re very close to our final result, which is Heisenberg
uncertainty principle formulated in , but to have a good intuitive idea of why it works, we need
to introduce one last thing: root mean square error. What is it? Practically speaking, it’s
the uncertainty of the measurement of a physical quantity. For an example, if the root mean square
error is very low, our precision will be high. To have a very simplified idea: if a Fourier
transform representing momentum is broader, the error regarding its value will be bigger.
Usually, physicists work their entire life on building appropriate devices that
reduce this error in order to obtain better results in their experiments.
Hundreds of books have been written on this topic, but when it comes to quantum mechanics, it’s
not possible to use tricks to reduce the error: there’s an inferior limit under which we can’t go.
Why? Because if we are measuring the parameters of a particle and we deal with its wave trying
to make a narrower spatial distribution, the relative momentum distribution (that is
the Fourier transform) will become broader and viceversa. This is the content of Heisenberg’s
uncertainty principle, which is one of the most important postulates of Quantum Mechanics.
We derived it from not rigorous wavy considerations, but its true formulation comes
from more advanced mathematical theories that we won’t examine. Just to be more precise here
is the rigorous definition: “the product of the uncertainty in the position and the momentum must
be higher than half the reduced Planck constant” (which is the planck constant divided twice pi).
So, if we try to reduce the root mean square of position to zero, understanding where the
particle is, the uncertainty on the momentum will grow rapidly to infinite and vice versa.
This result is at the same time extraordinary and terrifying! It’s totally a different way
of thinking about physics: nature itself, which can be considered as made by quantums
that are both particles and waves, prohibits us to measure everything with infinite precision.
We can’t calculate the exact trajectory of a particle. We must limit to say that there’s
a certain probability to find an object in a region: it’s the death of Newtonian’s determinism!
Yes, science is really powerful, but it is science itself that has discovered that not
everything can be predicted using maths and we can only limit ourselves to calculating probabilities:
something we can certainly manipulate well, but also something that is not
so distant from the Greek fate we were talking about at the beginning.
Are you amazed by quantum mechanics? Did you already know Heisenberg uncertainty principle?
Do you want to know more about wave-particle duality? Have you ever heard about Fourier?