Euler angles. Quantum Mechanics Question

AI Thread Summary
The discussion revolves around the relationship between the operators G_k and angular momentum operators J_i in the context of Euler angles and quantum mechanics. A participant questions the solution manual's assertion that G_i = J_i / ħ, suggesting it should be G_i = J_i * ħ instead. Another participant clarifies that G_i = -J_i / ħ is derived from the commutation relations resulting from infinitesimal rotations, leading to the conclusion that [G_i, G_j] = -iε_ijk G_k. The conversation emphasizes the importance of understanding the conventions used in quantum mechanics and the derivation of these relations from fundamental principles. Overall, the discussion highlights the nuances in operator definitions and their implications in quantum mechanics.
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Homework Statement



Let

U = e^{iG_{3}\alpha}e^{iG_{2}\beta}e^{iG_{3}\gamma}

where ( \alpha, \beta, \gamma ) are the Eulerian angles. In order that U represent a rotation ( \alpha, \beta, \gamma ) , what are the commutation rules satisfied by the G_{k} ?? Relate G to the angular momentum operators.

Homework Equations


The Attempt at a Solution



I attached here the solution that i saw in my solution manual.. My question is how did he get

G_{i} = \frac{J_{i}}{\hbar}

I think it should be G_{i} = {J_{i}}{\hbar}

Can someone please help me understand that solution?? Thanks. Help much appreciated.
 

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it should be one over h bar, and it is arisen from the commutation relations that result from infinitesimal rotations in 3 dimensional space. for more details please refer to advanced quantum mechanics texts as the proof is rather lengthy. i recommend reading:
http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf
specifically page 4, equations 9 to 12.
 
hi there. i have been working on this problem recently, but i seem to have a slightly different answer to the one above. my working out led me to have a minus sign in the relation between G and J:

after taking the taylor expansion of the exponentials and relating the \epsilon^{2} coefficients i got:

i^{2}G_{1}G_{2}-i^{2}G_{2}G_{1}=iG_{3}
so that [G_{2}, G_{1}]=iG_{3}
or [G_{1}, G_{2}]=-iG_{3}

this gave me [G_{i}, G_{j}]=-i\epsilon_{ijk}G_{k}

and hence i came up with the relation

G_{i}=-J_{i}/\hbar

is this difference just some use of a different convention, or am i doing something wrong along the way??

cheers
 
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