Graduate Integration with Euler angle of rotation matrixes

[RiceSweet]

Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write

in terms those three Euler angles.

 

[Svein]

There is something fishy here. Both your matrices have determinant 0!

 

[RiceSweet]

Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write View attachment 234555 in terms those three Euler angles.

View attachment 234556

Sorry that I made a mistake for one of mine matrix. The matrix of Sz should like as follows

 

[RiceSweet]

There is something fishy here. Both your matrices have determinant 0!

Hello, thanks for your reply. I made a mistake of Sz matrix, and I just updated the correct matrix. Those two matrices do have a determine as well as trace equal to 0 because those matrices are the matrix representation of Sz and S^2 operator of spin 1/2 1/2 system in Quantum Mechanics. The three exponential inside the integral represents the rotation.

 

[Svein]

So your rotations are non-invertible?

 

[RiceSweet]

So your rotations are non-invertible?

Yes, because those rotations are acting on spins. For example, if We act this rotation operator related to Sz on an eigenstate |00>, we will get e^(0) = 1 as our result, and we can not reverse our process. Since our eigenvalue of |00> corresponding to the Sz operator is just 0.

 

[RiceSweet]

So your rotations are non-invertible?

I have the general expansion form of the rotation operator already. The problem that I am struggling with is that I didn't know how to break down the derivative of this Euler angle vector (theta), and do the right integral.

 

[Svein]

Mathematically your expression does not make sense.

Since I never studied quantum mechanics, I cannot comment on whatever goes on backstage. I have, however, spent a couple of years working on robot arm coordinate transforms and I am therefore fully conversant with coordinate transform matrices.