Trying to learn the uncertainty principle

[mjc456]

I am trying to really, truly understand the uncertainty principle. I have done a lot of reading and self-study and listening to lectures online. I feel I'm getting closer, but would appreciate some help in directing my further efforts. Here is my horrible patchwork understanding thus far, which is no doubt riddled with misunderstandings and inaccuracies:
At some point we start calling measurements "operators". (The exact reason/justification for this, I'm not sure of). Like the "momentum" operator is what you use to calculate momentum. We start with a particle's wave function, which (I think) represents the probability distribution of the energy of the particle. For some reason real numbers aren't sufficient for this job, so complex numbers are involved in this (not sure why, and would like to find out). Anyway, we have this wave function, we apply an operator, say the "momentum operator", which is just a mathematical expression that we work on the whole wave function, and this transforms our old energy wave function into a new wave, which is also a complex function. This new wave maps the probability distribution of certain momentums.

The thing is that the "position" operator and the "momentum" operator don't commute, which means, the answers you get change when you reverse the order. This is much like matrix multiplication, in which (given matrices A and B) AB is not equal to BA. In fact, sure enough, it seems there are a bunch of linear algebra terms all through QM, but for QM what exactly there is that is supposed to comprise the elements of those matrices, I don't know. What do the matrices and their elements represent? How are they related to operators? And why do I need a Hilbert space?

The minimum amount of discrepancy between going Position then Momentum vs. Momentum then Position seems to be related to Planck's constant, and since the discrepancy has a constant minimum, voila, uncertainty principle.
Ok. So that was it. I would appreciate any help, either in the form of direct explanation, or even just some useful hyperlinks or books that I can read myself which would definitely address these issues. I'm not afraid to do the work myself, but I'm afraid I'll drown in information overload trying to just continue brute-force googling my way through the whole thing. Thanks for any consideration.

Also, I hope I posted this in the correct forum, if not I apologize and please let me know.

 

[sweet springs]

Hi.
The post seemed to me that you surely have eagerness to study QM.
I would like to recommend you one of may favorite books of a classic work of QM
https://www.amazon.com/dp/0198520115/?tag=pfamazon01-20
I was surprised to see there such expensive price. You may need some courage to purchase but it surely pays back. I hope you too like it.
Regards.

 

[zonde]

At some point we start calling measurements "operators". (The exact reason/justification for this, I'm not sure of).

I would say that it is the point where you move from real space into Hilbert space and represent ensemble using state vectors.

We start with a particle's wave function, which (I think) represents the probability distribution of the energy of the particle. For some reason real numbers aren't sufficient for this job, so complex numbers are involved in this (not sure why, and would like to find out).

It might be useful to look at http://en.wikipedia.org/wiki/Fourier_transform" . I guess it's because you don't have "real" (or do not have at all) time dimension when moving to Hilbert space.

Anyway, we have this wave function, we apply an operator, say the "momentum operator", which is just a mathematical expression that we work on the whole wave function, and this transforms our old energy wave function into a new wave, which is also a complex function. This new wave maps the probability distribution of certain momentums.

The thing is that the "position" operator and the "momentum" operator don't commute, which means, the answers you get change when you reverse the order. This is much like matrix multiplication, in which (given matrices A and B) AB is not equal to BA. In fact, sure enough, it seems there is a bunch of linear algebra terms all through QM, but for QM what exactly there is that is supposed to comprise the elements of those matrices, I don't know. What do the matrices and their elements represent? How are they related to operators? And why do I need a Hilbert space?

The minimum amount of discrepancy between going Position then Momentum vs. Momentum then Position seems to be related to Planck's constant, and since the discrepancy has a constant minimum, voila, uncertainty principle.

Have not come across nice experiments demonstrating Position/Momentum uncertainty. So I am hesitant to comment about that.
It's easier to think of experiments demonstrating uncertainty in polarization or spin measurements.