Jonathan Gleason
Reputation
2,715
Next privilege 3,000 Rep.
 Nov 9 comment Derivation of the irreducible representations of SO(3) Also, it cannot be related to $L_i$ at all. $L_i$ is an element of the Lie algebra, and we know that $\mathfrak{su}(2)=\mathfrak{so}(3)$. Nov 9 comment Derivation of the irreducible representations of SO(3) This is exactly what I am suggesting. This has nothing to do with spin vs. orbital. If somebody has told you that spin in $SO(3)$ can change a sign after a rotation by $2\pi$, they were wrong. This can only happen for spin with $SU(2)$. The point is is that, in quantum mechanics, we don't really care about representations, but rather projective representations (basically representations up to phase). It turns out that projective representations of $SO(3)$ are in canonical one-to-one correspondence with actual representations of $SU(2)$. Nov 9 comment Derivation of the irreducible representations of SO(3) Suppose that the representation has spin $n/2$ with $n\in \mathbb{Z}^+$. Then, in particular, this representation will contain an element $v$ such that $S_zv=\frac{n}{2}v$. This is at the level of the Lie algebra. At the level of the Lie group, $R_z(\theta )=\exp (-i\theta S_z)$ is a rotation about the $z$-axis by an angle $\theta$. Hence, $R_z(\theta )v=\exp (-i\theta \frac{n}{2})$. For $\theta =2\pi$, we had better have that $R_z(2\pi )=R_z(0)$; however, using the formula above, $R_z(2\pi )=\exp (-i\pi)=-1$. Thus, this does not give us a representation of $SO(3)$ for half-integer spin. Nov 9 revised Derivation of the irreducible representations of SO(3) added 19 characters in body Nov 9 comment Derivation of the irreducible representations of SO(3) The spherical harmonics give you all the irreducible representations of $SO(3)$, but it does not give you all those of $SU(2)$. You'll note this does not make any direct reference to the Lie algebra of $SO(3)$ Nov 9 answered Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one Nov 9 answered Calculation mistake some place in finding stress-energy tensor Nov 9 revised Calculation mistake some place in finding stress-energy tensor added 10 characters in body Nov 9 answered Derivation of the irreducible representations of SO(3) Nov 9 answered Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring? Nov 8 answered Neutrino-Neutron Interaction Feynman Diagram (W Boson Direction) Oct 21 awarded Notable Question Oct 15 awarded Nice Question Sep 30 awarded Explainer Sep 28 revised What are great circles of 2-sphere? deleted 2 characters in body Sep 27 answered What are great circles of 2-sphere? Jul 23 awarded Good Question Jul 22 awarded Famous Question Jul 2 awarded Curious Jun 11 revised The relation between classical and quantum vacua added 36 characters in body