" Subatomic particles do not just sit around being subatomic
particles. " - Gary Zuka
The Observable Universe which is assumed to be 46.508 billion light
years in size is fundamentally made up of 12 particles. Fermions, Bosons,
Quarks, Leptons and Anti-particles (there are some hypothetical particles
also), more on this later. The particles have a very strange concept with their position. They tend
to be nowhere until we find them everywhere.
Lets understand it with an example of Alex (an ordinary physics boy)
who is in a house of five rooms. Our task is to find his location
without entering in any of the rooms. Practically it seems absurd but
mathematically, we can guess that Alex is probably in any one of the
five rooms (1/5). Now lets dive into the Quantum Mechanical view of finding
the particle in space.
We will stick to Copenhagen Interpretation as it gives a more clear
description while being widely accepted among the physicists. In simple
words, it states that psi ψ has infinite number of superposition in space. And as soon as we
observe our wave function, it collapses to one giving an arbitrary
solution.
We will talk, as in the case of Alex, in probabilities to locate our
particle in space* as the particle is localized at a point but it's
wave function is spread across the space. It seems absurd at first but
then the Born's Statistical Interpretation |ψ(x, t)|2 enters to fix the absurdity. It basically finds all the probabilities of
the wave function at a specific position with respect to a specific
time.
In a more rigorous way, we can go within a big range, as in Probability
Density from a to b (it can be assume to be from -infinity to +infinity)
with respect to time.
\int_{a}^{b}\left | \psi(x,t) \right |^2 dx
The equation was formulated by Max Born in 1926 which in its simplest
form states that the Probability Density of finding a particle at a given point, when measured, is
proportional to the square of the magnitude of the particle's wave
function at that point.[1] The Probability Density is the tool which finds the most probable area for the wave function to be found in space.
It is said that Erwin Schrodinger presented an equation that treated the electron as a
wave, and Born discovered that the way to successfully interpret the wave
function that appeared in the Schrodinger equation was as a tool for
calculating probabilities
[2].
Till now the particle was in space but what happens after we observe it's
wave function. Interestingly it collapses to one as it won't make sense if
i say that Alex is present in all the five rooms even after we have
spotted him in the first room. Though in Quantum Mechanics, sometimes we have to normalize the
wave function to one.
\int_{a}^{b}\left | \psi(x,t) \right |^2 dx = 1
If you are very curious then at first you might wonder as where was the
particle before it was found, was the particle's position already fixed or
was it given. Same thing was asked in orthodox position but the Copenhagen Interpretation throws the concept of
superposition which make the question meaningless.
After that, we apply Schrodinger equation to our normalized wave function which makes the wave
function independent of time thus keeping it normalized
forever.
I think till now it's understood that everything, starting from observation to preserving our wave function, is done by us. The funny part is that till this time no one has a clear definition of how to interpret this wave function 𝛙 and the amazing part is, even after this divergence in interpreting the wave function, over the years, we have made progress in Quantum Mechanics.
In conclusion, a particle is nothing without it's normalized wave
function. Even though one can argue on the deterministic nature of
classical physics but entering in the world with a scale, which is so
minute that one can only work in probabilities albeit making it Indeterministic. The particle have a
strange nature, which in fact is fascinating to imagine as they will still be in a probabilistic location which mathematically can be preserved
but in reality, changes every fraction of a second.
Notes-
* As particles exist in infinite superposition of
itself even at a single point therefore we can't have an exact number for putting up a specific
range.
** A wave function |Ψ| is a complex function which tells the mathematical
description of a particle in any of the spatial dimensions like
|Ψ(x)|2 for the wave function with respect to
position but independent of time.
A wave function doesn't have any physical Interpretation, so don't be confused with it's operations and alone it does not represent the probability, but a
probability amplitude.
References-
[1] https://en.wikipedia.org/wiki/Copenhagen_interpretation#Probabilities_via_the_Born_rule
[2] Bernstein, Jeremy (2005). "Max Born and the Quantum Theory". American
Journal of Physics. 73(11): 999–1008. Bibcode:2005AmJPh..73..999B. doi:10.1119/1.2060717
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