Euler angles. Quantum Mechanics Question

[jhosamelly]

Homework Statement

Let

U = [itex]e^{iG_{3}\alpha}[/itex][itex]e^{iG_{2}\beta}[/itex][itex]e^{iG_{3}\gamma}[/itex]

where ( [itex] \alpha, \beta, \gamma [/itex] ) are the Eulerian angles. In order that U represent a rotation ( [itex] \alpha, \beta, \gamma [/itex] ) , what are the commutation rules satisfied by the [itex] G_{k} [/itex] ?? 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

[itex]G_{i}[/itex] = [itex]\frac{J_{i}}{\hbar}[/itex]

I think it should be [itex]G_{i}[/itex] = [itex]{J_{i}}{\hbar}[/itex]

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 [itex]\epsilon^{2}[/itex] coefficients i got:

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

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

and hence i came up with the relation

[itex]G_{i}=-J_{i}/\hbar[/itex]

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

cheers