Euler angles. Quantum Mechanics Question

[jhosamelly]

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.

 

[ardie]

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.

 

[beans73]

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