Wait is that not true? Why wouldn't H form a right angle with P and A?
AH would be perpendicular to n, and PH would be parallel to n, making them perpendicular to each other? Or am I misunderstanding the definition of a plane projection?
The position-momentum uncertainty relationship is just a specific case of a more general relationship. There are other uncertainty relationships, such as between time and energy or between two (separate/orthogonal) components of angular velocity. The relationships basically state that whenever you measure one of the two values, you are required to add uncertainty to the other.
Unfortunately, this is kinda where my knowledge on the subject starts to hit its limits. As for spin, it has a lot of effects on the energy of the system it's involved with, so I believe the energy-time or angular momentum exclusion principles would apply there.
You might also be thinking "why not have two entagled cloned particles, and measure the momentum of one and the position on the other?". While you can duplicate particles, there are reasons why that doesn't work that I don't really remember tbh. I'm sure PBS Spacetime on Youtube has an episode on it somewhere though if you're interested
What I mean to say is that the detector is not what's changing the particle; It's the process of learning about an aspect of the quantum system that forces it into one state or another (at least from our own personal perspectives).
Sorta! According to the Heisenberg Uncertainty Principle, there's an upper limit to how much we can "know" about the given state of a quantum system. This isn't an issue with our measurements, but a fundamental property of the universe itself. By measuring one aspect of a quantum system (for example, the momentum of a particle), we become less certain about other aspects of the system, even if we had already measured them before (such as the position of the same particle).
Though (as far as we know), we aren't going to run out of quantum states or anything like that.
Not exactly. Quantum physics applies no matter how you measure it. The double-slit experiment is an example of this: Photons moving through two slits will form a wave interference pattern on a detector plate, even though the detector doesn't affect the position of the photons beforehand.
It's more like: when you become aware of the results of a quantum measurement, you yourself become a part of the quantum system, and being a part of the system requires measurements to have real values. Whether you should interpret this as a wave-function collapse or branching into multiple parallel universes is up for debate though.
I've actually messed with this a bit. The problem is more that it can't count to begin with. If you ask it to spell out each letter individually (ie each letter will be its own token), it still gets the count wrong.
Well from a hardware perspective, pretty much every tech product is built on the back of horrifying amounts of labor exploitation.
Also in some cases components will contain gelatin, which isn't vegan.
For software? Well I sacrifice and goat and feed it to the machine before every git commit I make, so that's probably not vegan either.