Clarifying the Superposition Principle

[MatinSAR]

TL;DR

Clarifying the Superposition Principle: Writing |ψ⟩ as a Linear Combination of Possible States.

Hi everyone,
I’m currently learning quantum mechanics and trying to wrap my head around superposition, measurement, and how repeated observations work. Here’s how I understand it—please let me know if I’m on the right track or if there’s anything I might be missing!

I’m trying to understand how different wave functions (ψ) like $ψ_1$ and $ψ_2$ in above picture, relate to probabilities in a quantum system.

Do different ψ's (like ψ₁, ψ₂, in above picture) represent entirely distinct probability distributions? For example, does ψ₁ predict a high probability of finding the particle at position x=2, while ψ₂ predicts it at x=3?
Or ...
Is there a single probability distribution described by one ψ, where ψ(x₁), ψ(x₂), etc., give the probabilities for different positions (x₁, x₂, ...) within that same distribution?
In other words, is each ψ its own 'probability rule' for the system?

 

[PeroK]

The wave function itself is a distribution of probability amplitudes. The modulus squared gives a probability distribution.

That's why a superposition of wavefunction involves interference between the wavefunctions.

 

[MatinSAR]

The wave function itself is a distribution of probability amplitudes. The modulus squared gives a probability distribution.

That's why a superposition of wavefunction involves interference between the wavefunctions.

I still don't fully understand what $\psi_1 (\vec r,t)$ and $\psi_2(\vec r,t)$ represent. Do they describe different distributions across the entire domain, or does their square represent the probability of finding the particle at a specific position $\vec r$?

 

[PeroK]

I still don't fully understand what $\psi_1 (\vec r,t)$ and $\psi_2(\vec r,t)$ represent. Do they describe different distributions across the entire domain, or does their square represent the probability of finding the particle at a specific position $\vec r$?

Each is a valid wavefunction. And a normalised linear combination of the two is a valid wavefunction.

Are you familiar with the superposition of electric fields? It's the same idea.

There's nothing mathematically radical about quantum superposition.

 

[MatinSAR]

Each is a valid wavefunction. And a normalised linear combination of the two is a valid wavefunction.

Are you familiar with the superposition of electric fields? It's the same idea.

There's nothing mathematically radical about quantum superposition.

So my first understanding in post #1 was correct.

Do different ψ's (like ψ₁, ψ₂, in above picture) represent entirely distinct probability distributions? For example, does ψ₁ predict a high probability of finding the particle at position x=2, while ψ₂ predicts it at x=3?

In this context how the coefficients determined? By repeating same experiment many times?

Are you familiar with the superposition of electric fields? It's the same idea.

Not sure But I will check my books.

 

[PeroK]

In this context how the coefficients determined? By repeating same experiment many times?

Yes. In terms of experimentally confirming a given procedure produces a given superposition of wavefunctions.

 

[MatinSAR]

Yes. In terms of experimentally confirming a given procedure produces a given superposition of wavefunctions.

Thank you for your help @PeroK
Comparing the textbook content with these ideas requires more time.

 

[bhobba]

Thank you for your help @PeroK
Comparing the textbook content with these ideas requires more time.

Indeed.

In QM, one builds up understanding over time by reading more advanced material. Eventually, you should read Ballentine—QM: A Modern Development. Don't worry at first; the nuanced understanding will come with time.

Thanks
Bill

 

[MatinSAR]

Indeed.

In QM, one builds up understanding over time by reading more advanced material. Eventually, you should read Ballentine—QM: A Modern Development. Don't worry at first; the nuanced understanding will come with time.

Thanks
Bill

Appreciate the tip! I’ll definitely check out Ballentine once I’ve leveled up a bit more.

Thanks for the reassurance!