normally we treat the associated legendre functions as polynomials that satisfies $\int_{-1}^1P^m_aP^m_b=C(m,a)\delta_{a,b}$

oh wow just as i wanted to find the solution to post XD thanks for the solution though, though still no clue what landau intended mdr

You may want to look at Kepler's problem solution(en.wikipedia.org/wiki/…), any 2 body system, with mass $m$ and $M$ can be reduced to a fixed mass of $M+m$ and a orbiting mass of $\frac{Mm}{M+m}$, the reduced mass. Such precession is indeed observed relativistically, but for a simple inverse square force it's likely numerical error

$F=\frac{dp}{dt}=ma+v\frac{dm}{dt}$, $\frac{dF}{dt}=mj+2a\frac{dm}{dt}+v\frac{d^2m}{dt^2}$

assuming mass is constant

octet is basically comes from energy bands. Pauli exclusion is not empirical, it was derived for fermions of half integer spin

If the spring has mass, does it lose energy as heat when being contracted or absorb heat when it contracts? Yes, which means energy is being exchanged with the environment. It's similar as considering a block moving from point A to B with friction. Some energy also goes into rearranging the lattice in the spring, which is also dependent on time

The question is how to we represent A. A has a value of $\frac{1}{2\pi\hbar}$ along with units that comes from the delta function

Good point, in the short section I've presented, many many justification steps are ignored. However, the wave function still should have units of $m^{-\frac{1}{2}}$ and total probablity of measuring a particle at any position is 1. P.S. really the momentum eigenvectors are just the position being completely undefined

If that isn't one there are massive implications on quantum mechanics!

Yeah I was just asking for notation. And yeah it shouldve been root not inverse root