I hope this is the right forum, this is mostly about maths, I'm not looking for a physical interpretation of angular momentum... yet. It also involves *some* calc... anyway... OK, firstly, I've come to the conclusion I don't get cross products. I understand the properties of them, and can use them OK, there's just something I came across that I don't get. Say you have [itex]\vec{a}\times\vec{b}=\vec{c}[/itex]. Apparently, the magnitude of c is given by the area of the parallelogram formed by a and b. I'm ok with the product axb having units of area, but when you then go and say c has a length that is an area.... I get a bit lost. How am I supposed to interpret that? Actually, scrap the second part, I'm an idiot! Oh, and my lecturer always seemed to swap between J and L, both apparently for angular momentum. They mean the same thing, right? Or have I completely not understood anything?? I'll be back. I hate angular momentum.
This kinda looks pointless now, I should always think through a problem thoroughly before looking for help on here! I still don't get the first bit about cross products though.
J is the 'total' angular momentum, which is a coupling of the orbital angular momentum and the spin angular momentum. http://en.wikipedia.org/wiki/Angular_momentum_coupling#LS_coupling http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html As for the cross product - http://hyperphysics.phy-astr.gsu.edu/hbase/vvec.html If vrectors a,b had dimensions of length, then in a x b = c, c would have magnitude of area, and vector would be parallel to the normal of the area. When we do v x B for the Lorentz force, the resulting vector has units of T-m/s, which have to be equivalent to N/C, since F = q(v x B). See also - http://en.wikipedia.org/wiki/Cross_product#Geometric_meaning
Think about what quantities angular momentum contains, i.e., what information is the angular momentum vector composed of; the scalar quantity - mass, the vector quantities - velocity and position. Isn't it a intuitive need to know what mass a particle has, where it is located, and what it's velocity is? Regarding the area/length affare - I wouldn't loose my head thinking about that too much if I were you. The vector c = a x b can have any physical meaning, so it's dimension can be length, velocity, acceleration, force, etc. It's absolute value always equals the area of the a x b paralelogram, but that doesn't mean the dimensions equal, too.
Thanks guys. I've not done any QM yet, maybe my lecturer was crossing between J and L subconciously. I think I see the connection, sort of! I can accept the thing about the magnitude of the cross product too, it kinda caught me off guard! Thanks again. :D