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