I was talking to a family friend in the field of optics at a quantum scale (not sure the proper name for this) and he was explaining to me why you can only determine either the velocity or location of a particle but not the other. He watered it down for me and told me this: You have an initial equation describing the particle (maybe the Schrodinger equation, but I'm not sure) and the Fourier transform. He said the Fourier transform is an operation. You can use the Fourier transform turn the equation into a new one that describes the velocity precisely, but the position less precise, or you can use it to find the location precisely, but the velocity imprecisely. He said it is irreversible so once you do the transform, you can't go back. My question is: why can't I copy the initial equation down on another piece of paper and give it to somebody else to find the velocity while I determine the location?
The no cloning theorem (which follows directly from linearity of time evolution) says you can't make a copy so you can't do that.
But that is not the point. The point is that you can make a state and even of you made a thousands states just like, the state itself will not give good results for position and good results for velocity.
There are states that give good results for position and states that give good results for velocity. But one that is really good at one is bad at the other.
So when you pick a state to have lots of you can pick one that is good at position or pick one that is good at velocity.
So why can a state be good at one and not the other?
First, keep in mind that when you get a position or a velocity you generally change the state. And position and velocity are things where the order you do them always matters. So the states you turn something into when you get position must be different than the thing you turn it into when you velocity (otherwise if it was a state that was a final state for each then neither would change it and the order wouldn't matter for that state but for positions and momentum the order always matters). So this indicates that when you measure position and then velocity and so on that you are getting different results. So a good position gives in good (can be different) results for velocity and vice versa.
So you might then ask why position and velocity have to have the order matter. Some people actually say they use that fact as the basis for getting all of quantum mechanical (technically it isn't velocity it is momentum they use and even more technically it is canonical momentum they use, not kinetic momentum). So some people will stop there.
But they aren't being totally honest. That would be valid if they got everything from that, but they do actually have to generalize it to get everything they have to generalize it to a general Poisson brackets go to an imaginary times a commutator. And just one of those is enough. Having the time evolution is enough to explain everything that happens in time.
So a true and proper explanation would have to explain how you get the results over time. Bit it doesn't end up being different, over time you get the initial state to split as the different options become entangled with the different states of the measurement device and they do it with sizes of the branches that lead to certain frequencies of results as measured by separate devices that measure frequencies.
And then the frequencies follow the same rules that position and momentum can have a small spread of values for one of them or the other one can have a small spread of values. And since the order you do them (now a dynamical consequence of the dynamics of how you actually measure them experimentally) matters for every state then again this is forced upon us. And if you want to compute the spread of values you get from repeated interactions, then you can find the spread of one is given by the standard deviation of a function and the spread of the other is given by the standard deviation of the Fourier transform of the first function
But those equations are just equations, nature doesn't pass along copies of the equations it just dynamically evolves.
You cannot simultaneously determine the position and the momentum of a particle for exactly the same reason that you cannot simultaneously determine what I am eating and what I am jumping over --- namely, I never both eat and jump at the same time, and a quantum particle never has both a location and a momentum at the same time. (Usually, it has neither.) You can force me to eat, and observe my food intake, but then I won't be jumping. You can force me to jump, and observe what I jump over, but then I won't be eating.