I was just listening to Richard Feynman’s “Six Easy Pieces” lectures (god I love mp3 file sharing). Specifically, I was listening to the fourth lecture, “Conservation of Energy.” Towards the end, he mentions that the Law of Conservation of Energy is related to the phenomenon of the invariance of physical laws with respect to time. Then he mentions that the Law of Conservation of Momentum is related to the phenomenon of the invariance of the physical laws with respect to position, and the Law of Conservation of Angular Momentum with respect to rotation. OK, methinks, so I know of two quantum uncertainty laws – one relates uncertainty in energy with time, and the other relates uncertainty in momentum with position. This leads me to wonder if there is a third quantum uncertainty relation – one between angular momentum and the rotational position of a particle. And if so, does this have anything to do with why the quantum property of “spin” is so odd, and dissimilar to its macroscopic rotational equivalent?
Also, Feynman mentioned three further conservation laws: Charge, Baryon Number, and Lepton Number. Do these laws have uncertainty relationships? Is there an uncertainty in the charge of a system relative to its […], for instance? Do virtual particles appear with unbalanced charge, but pay for it in some other quantity (analagous to virtual particles that violate Conservation of Energy, but can only exist for short times)? These lectures were recorded in 1961, so are these last three conservation laws even accurate any more? Are there others? I vaguely remember reading some Asimov that I believe mentioned some others…Conservation of Strangeness, perhaps, or Conservation of Color?
Have a look at a good book on mechancs, like Symons, and read the sections on the Lagrangian, for instance. If the angular positio is not explicit in the Lagrangian, then the motion is symmetric in rotation and angular momentum is conserved. It’s similar for linear momentum. There must be some sort of similar relationship connecting energy and time, but I don’t know what it is. And all of this is classical mechanics.
It’s a reltionship between symmetry and conservation, an has nothing to do with uncertainty.
When it comes to quantum mechanics and uncertainty relationships, then the relationship mentioned by Achernar is true – operators that don’t commute represent quantities that cannot be simultaneously measured to arbitrary precision, like linear momentum and position. In other words, xp(psi) - px(psi) isn’t zero, where Psi is your wavefunction, x is the position, and p is the momentum.
“You mean a times b isn’t the same as b times a?”
“Not by the time these babies get through with them.”
– Heinlein, in one of his books. “Fifth Column”, I think. This is the fis time encountered a case where this was true.
Is this a standard uncertainty relationship? I.e., if you don’t precisely know the vector length and z component, does that increase your knowledge of the x and y components? Or are length and z what you will always know, and x and y what you can never know? For instance, could you know the xy plane of the spin precisely, but have no knowledge of the vector length or z component (I think this would give you the equatorial plane of rotation, but leave you in the dark about spin speed and direction)?
Also, does this account for the unusual properties of quantum spin? I.e., that particles have discrete values for spin (the quantified vector length) and are spin up or spin down (positive or negative z), but that’s about it?
I am still unclear whether uncertainty is supposed to have ontological importance or not. So many times I here it is inherent in these quantities, so many others I hear that it comes from the act of interfering in measuring with linked qualities.
You can choose any axis you like, and find the angular momentum about that axis, but then you can’t learn the angular momentum about the other axes. It’s just that physicists usually call the axis that the know the z axis (might as well; there’s no better definition of the z axis out there). Note also that the uncertainty relation is a lower bound: Just because you increase your uncertainty of one component, doesn’t necessarily mean that you learn more about the others. It just means that you can learn more about the others.
As for symmetries and conservation laws, if the Lagrangian does not explicitly depend on time, then the Hamiltonian is conserved. The Hamiltonian is usually, but not always, the same thing as the total energy. All conservation laws can be related to symmetries, and vice versa, but the relationship is not necessarily obvious: For instance, charge conservation is tied to something called gauge symmetry, which refers to a mathematical freedom in how you can describe electromagnetic fields.
