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seen Oct 10 '13 at 12:09

Jun
29
comment Clebsch-Gordan Identity
@Dan: The sums are over all valid values of the arguments, specifically $-a\leq\alpha\leq a, -b\leq\beta\leq b, -d\leq\delta\leq d$
Jun
28
asked Clebsch-Gordan Identity
May
26
comment Why are two eigen-state-kets with different eigenvalues orthogonal?
yes, I was making this mistake. Thank you for clarifying, I think part of the confusion was the idea that the indices can take non-positive values (i.e. for $j_1=1$, $m_1$ can take any of $\{-1,0,1\}$ so $\vec{e}_m$ can be any of $\{ \vec{e}_{-1}, \vec{e}_0, \vec{e}_1 \}$. All of which are mutually orthogonal. Am I understanding this correctly now?
May
26
comment Why are two eigen-state-kets with different eigenvalues orthogonal?
@Peter: so I assume the proof would be something like: define $\left|j_1 j_2 m_1 m_2\right> = \sum_j \sum_m \left<j_1 j_2 m_1 m_2 \mid j_1 j_2 j m\right> \left|j_1 j_2 j m\right>$, where $\left<j_1 j_2 m_1 m_2 \mid j_1 j_2 j m\right>$ are the Clebsch-Gordan coefficients. Do the same thing for a different ket, say, $\left|j_1' j_2 m_1' m_2\right>$, then their inner-product is the sum of the products of their coefficients, which should be zero (I assume there will be some kronecker deltas in there somewhere too).
May
26
comment Why are two eigen-state-kets with different eigenvalues orthogonal?
@Peter: thank you for your explanation that was very helpful. So if I'm understanding you correctly the dual bra to a given eigenket (in my case a simultaneous eigenket) is defined to give 0 as their inner-product. If this is correct, then in practical terms, given some numerical values for $j_1, j_2, m_1, m_2, j, m$, how do I compute the inner product? (i.e. the definition we've imposed tells me what the inner-product of two orthogonal kets is, but it tells me nothing about what the inner-product of two general kets is so that I can check if they are orthogonal).
May
26
comment Why are two eigen-state-kets with different eigenvalues orthogonal?
I admit I'm a beginner and probably am making A LOT of beginner's mistakes. I know the derivation is incorrect, I want to know what I did wrong so that I can correctly understand.
May
26
asked Why are two eigen-state-kets with different eigenvalues orthogonal?
May
25
accepted state vector notation
May
24
comment state vector notation
@David: thank you, your explanation was very illuminating. Perhaps I will chat about this a bit later, it would definitely be beneficial for me. I think armed with your explanation I'm going to try to read some more from Sakurai's Modern Quantum Mechanics.
May
24
comment state vector notation
@Lubos Motl: So is this to say that in a manner analogous to the similarity transformation $B=P^{-1}AP$ the function of the Clebsch-Gordan coefficients is that of the matrix P, i.e. if A is a matrix whose columns are the basis kets for some hilbert space H1 and B is a matrix whose columns are the basis kets for some other hilbert space H2 then the similarity transformation (change of basis) is effected by using the matrix P whose elements are the Clebsch-Gordan coefficients?
May
24
comment state vector notation
@dbrane: so is there any difference between $\left|jm\right>$, $\left|j,m\right>$, $\left|j;m\right>$?
May
24
asked state vector notation
May
19
awarded  Scholar
May
19
accepted Integral of the product of three spherical harmonics
May
19
awarded  Supporter
May
18
awarded  Student
May
18
comment Integral of the product of three spherical harmonics
@Ramashalanka: Arfken mentions the identity but doesn't provide a derivation, what I'm interested in is the actual derivation.
May
18
asked Integral of the product of three spherical harmonics