August 19, 2023
In quantum mechanics, the properties of a particle are described using probability amplitude functions. The probability of a the particle taking a certain value is related to the output of the probability function at that value.
Here, we can see two probability functions, \( \color{#5492f0}{\psi(x)} \) and \( \color{#F09B24}{\psi(p)} \), representing probabilities for position (\(x\)) and momentum (\(p\)) respectively.
Because this function is complex valued, we are only plotting the real part of the wave here. The following demo shows the related probabilities, making for a more intuitive example.
The essence of the uncertainty principle is the fact that, the more precicely you narrow down the position (represented by the width of possible values it could be), the less precicely you know its momentum. That is if the probability of position is not spread out, the probabilities of different momentums must be, and vice versa. In the extreme cases, where position has a 100% chance of being some exact value, the momentum nescecarily spreads out to have an equal probability of being any value. Conversely, if you know with 100% certainty the momentum, the particle will have an equal chance to be found at any location in the universe.
An idea like this is very counterintuitive, and it begs the question of why something like this is true at all.
As we saw earlier, the probabilities of speed and momentum are represented with a wave. The key idea is the relationship between the two waves for position and momentum. To get an idea of how they are related, consider the following experiment.
You have a stationary wave, say a bent piece of wire on a table. Imagine you can only see the part of the wave directly in front of you. To get an idea of its frequency, you could "scan" the wave by walking past it, recording what you see as you move over time.
The faster you walk past it, the higher the frequency appears to be. In General, the frequency you see depends on the relative motion between you and the wave.
Its motion is therefore encoded in the frequency you observe. Meaning momentum, a measure of movement, is proportional to the frequency of this wave.
The final piece of this effect is to explain why the waves related in this way produce this effect. That I leave to 3Blue1Brown in this clip from his video (worth watching on its own as well). In this analogy the "time" is the width of our piece of wire on the table. and as we saw, the frequency we percieve is the momentum.