Wave or Particle : Uncertainty is Inevitable
Uncertainty is not "I don't know". It is "I can't know". "I am uncertain" does not mean "I could be certain". ~ Werner Heisenberg
Similarly, I created another Python simulation to demonstrate Heisenberg’s Uncertainty Principle (uncertainty in position) using wave superposition. The 1st graph in above picture shows the superposition of five plane waves having the same frequencies and amplitudes. The resultant wave in the 2nd graph has multiple peaks showing equal probabilities of finding the particle in that region (we mainly take |Ψ|² of the wavefunction)- hence uncertainty in the position. The wave in the 2nd graph is made up of different plane waves having the same frequency, so it has a definite frequency, hence certainty in the momentum.
The Classical World : We know that in classical physics, matter is considered to be made of discrete particles with definite positions and velocities. These particles follow deterministic laws, such as Newton’s laws of motion, meaning that their future behavior can be precisely predicted if initial conditions are known. Classical particles do not exhibit wave-like properties. The particle concept suffices to describe macroscopic objects, from planets to billiard balls, where quantum effects are negligible.
The Quantum World : The mysterious energies at the quantum scale (less than nanometers) show dual nature : particle nature and wave nature. The Photoelectric Effect is an experimental proof showing particle nature while the Young's Double Slit experiment showing the interference proves the wave nature. The Heisenberg's Uncertainty Principle emerges as a consequence of the wave particle duality.
Let us read below about the Heisenberg's Uncertainty Principle and how this principle rules in both particle as well as wave nature of quantas.
PARTICLE NATURE
In the above picture the tennis ball appears sharp and well-defined. We can tell from the picture the exact position and also the direction of momentum—that the ball is moving towards east, simultaneously. This is because quantum effects are negligible at large scales as the ball’s mass is large compared to fundamental quantum particles. As a result, we can track its trajectory, speed, and position simultaneously with great accuracy. So, according to classic mechanics, if we know the initial conditions of a macroscopic object, then we can predict with certainty its future and past conditions. This was true until Werner Heisenberg, one of the fathers of Quantum Mechanics, decided to leave us in an AWE !
The Heisenberg's Uncertainty Principle states that we can not know simultaneously, with certainty, the position and momentum of a particle. The position and momentum are canonically conjugate quantities in quantum mechanics. In quantum mechanics, canonically conjugate quantities are pairs of physical variables where knowing one precisely makes the other uncertain. This does not appear to be a result of a lack of technology but rather a fundamental characteristic of our cosmos. Interestingly enough, this not only applies to the position and momentum but also to all the canonically conjugate variable pairs, for example, time and energy.
Now, suppose the tennis ball is a microscopic/quantum particle like an electron, as shown in the picture below.
In picture 1 (above), we can predict the position of the ball where it appears sharp, but now can we precisely tell in which direction the ball is moving ? The answer is NO ! The ball could be moving in any direction. For this case, the position graph 2 is very narrow, meaning the particle’s position is well localized. But the corresponding momentum graph is broad, meaning its momentum is highly uncertain.
In picture 2 (above), we can predict that the quantum tennis ball is moving towards north-east, but can we precisely tell the position of it ? The answer is NO ! The position seems to be blurred. For this case, the position graph 1 is wide, meaning the particle’s position is uncertain and spread out. But the corresponding momentum graph is narrow, meaning its momentum is well defined.
Real Example : Electron Cloud
I created this Python simulation to visualize the electron cloud of the hydrogen atom. These plots show the probability density distribution for different quantum states (100, 200, 300, 400) in the XZ-plane. Light areas indicate regions where the electron is most likely to be found. Fascinating to see quantum mechanics come to life like this!
A clear real-world example of the uncertainty principle is the hydrogen atom. The electron is not found orbiting the nucleus in a fixed path, as in classical models, but instead exists in a “cloud-like” orbital. This cloud represents the probability distribution of where the electron might be. If the electron’s position near the nucleus were precisely known, its momentum would be highly uncertain, and vice versa. The orbital shapes (s, p, d, etc.) are direct consequences of this balance between position and momentum uncertainty, giving atoms their unique structure. This probabilistic picture of the electron is a direct manifestation of the Heisenberg Uncertainty Principle. Confining the electron too close to the nucleus (a very precise position) would require it to have a very large momentum, making it unstable. Instead, the electron occupies stable atomic orbitals where the uncertainty in position and momentum balances naturally. These quantized orbitals explain why atoms have discrete energy levels and discrete angular momentum, why they absorb and emit light at specific frequencies, and why chemical bonds form in the way they do.
WAVE NATURE
de Broglie proposed a hypothesis that matter, like light, exhibits a dual nature. All moving particles—whether they are macroscopic objects like you, me, or celestial bodies—or microscopic quantum particles like electrons and protons, etc.—have wave-like properties. For every moving particle, there is an associated wave called the matter wave with a wavelength called the de Broglie wavelength (λ). The de Broglie wavelength is inversely proportional to the momentum of the particle. Equivalently, the momentum is directly proportional to the wave's frequency. Now, a single de Broglie wave with a definite frequency (and hence definite momentum) is spatially infinite—it extends throughout all space. This raises a fundamental problem: if a particle is described by a single de Broglie wave, how can we know where the particle actually is, since the wave is spread out over infinite space? To resolve this, the concept of a "wave packet" was introduced. A wave packet is formed by the superposition of multiple waves with slightly different frequencies and wavelengths. Due to this superposition, interference occurs: at most points, the waves cancel each other out (destructive interference), but at other times, they reinforce each other (constructive interference). The wave packet describes a particle not as a single point but as a spread-out but localized (not infinite) wave.
In the above pic, the bottom left graph shows different waves having different frequencies, and the bottom right graph shows the resulting localized wave packet (superposition of waves). The region of maximum amplitude in the wave packet represents the region with the highest probability density of finding the particle. The overall envelope of the wave packet moves with a velocity called the group velocity, which corresponds to the velocity of the particle. The individual waves within the wave packet, known as phase waves, can move with different velocities known as phase velocity. In essence, the de Broglie wave packet is a mathematical construct that incorporates both wave-like and particle-like aspects of matter, offering a more accurate description of a particle’s motion in quantum mechanics. But here also the Heisenberg's Uncertainty Principle rules that we can not measure two canonically conjugate quantities like position and momentum of the wave simultaneously.
Python simulation I created to demonstrate Heisenberg’s Uncertainty Principle (uncertainty in momentum) using wave superposition. The 1st graph in above picture, shows the superposition of five plane waves having different frequencies and amplitudes. The 2nd graph shows a wave packet as a result of the superposition of all five plane waves shown in the 1st graph. The peaks in the wave packet show a high probability of finding the particle in that small region (we mainly take |Ψ|² of the wavefunction)- hence certainty in the position. As the wave packet is made up of plane waves having different frequencies, it doesn't have a definite frequency, hence uncertainty in the momentum of the particle.
Now the question arises - Why does this Uncertainty exist ?
The Quantum Balloon
This is the non-deterministic, probabilistic part of quantum mechanics. The measurement "forces" the particle into one of the available eigenstates of the measured observable. If we were to immediately measure the same property again, you would get the same result with 100% certainty because the state is no longer a superposition but is now in the particular measured eigenstate. This is the core of our description : the interaction to measure position collapses the wavefunction, leaving the particle in a definite position eigenstate but destroying all previous information about its momentum, because the position and momentum eigenstates are incompatible (non-commuting).
The Built-in Blur of Reality
The reality of the quantum particles is based on the Quantum Balloon concept. Suppose we want to know the position of a quantum particle accurately. Then we have to use a microscope having high resolution power. A microscope uses radiation. We know the shorter the wavelength of radiation used, the higher the resolution power. When this radiation or the photon is incident on the particle, there will be a two body collision between the photon and the particle. The momentum of photon can change the momentum of the particle when they collide. Consequently we cannot determine the momentum accurately.
On the other extreme if we want to determine the momentum of the particle accurately we have to use a radiation having lower momentum i.e. of longer wave length. In this case the resolution power of the microscope will be poor and we cannot determine the position of the particle accurately. Therefore it is quite clear that we cannot determine the position and momentum of the particle simultaneously with arbitrary precision.
This does not appear to be a result of a lack of technology but rather a fundamental characteristic of our cosmos. The above was for the 'particle' character. We can obtain the same conclusion from the 'wave' character.
Suppose the matter wave is a plane wave having definite momentum. If it is so, then its momentum is definite or uncertainty in momentum is zero. But the plane wave is completely delocalized and hence its position is completely arbitrary. In other words
the uncertainty of position tends to infinity.
On the other hand if we want to make its position definite, we have to take the super-position of an infinite number of plane waves having different momentum. Then it may happen that due to constructive and destructive interference the resulting wave may be peaked at a particular point that corresponds to its position. Therefore its position can be definite with zero uncertainty. But for this wave the momentum is completely arbitrary i.e. its uncertainty in momentum tends to infinity.
Restlessness of the Blurry Quantum World
Zero-point energy is the direct consequence of Heisenberg's Uncertainty Principle. The Zero-point energy is the lowest possible energy that a quantum mechanical system can have, even in its ground state (the state with the least energy). In classical mechanics, a system could theoretically be at rest (with zero energy), but quantum systems never quite "rest" due to quantum fluctuations. These fluctuations are a direct consequence of the uncertainty principle.
For a particle in a potential well (such as an electron in an Hydrogen atom), the uncertainty in position (Δx) means there is an uncertainty in momentum (Δp), even in the ground state. The uncertainty in momentum means the particle still has motion, and it can not be perfectly "still". Therefore, kinetic energy manifests as zero-point energy.
Henceforth, quantum mechanics dictates that the particle can not have zero energy because that would require precise knowledge of both its position and momentum which in turn would violate the uncertainty principle. This results in the particle not being able to have a classical "rest" state, where its energy would be zero.
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Feel free to comment your thoughts and curiosities and also point out mistakes if any.
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Thanks for Reading !
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