Euler/Tait-Bryan angles definition

[andreamarini]

In order to define the symmetry operations in spin space the first step is to evaluate the Euler angles (the image below is from Sakurai, Modern Quantum Mechanics 1994 edition, page 172).

The procedure is tricky as Sakurai discusses how the order of rotations must be carefully choosen (indeed he comments on the order reported in the Goldstein book on classical mechanics, stating that it is not consistent with QM).

In Yambo such angles are defined in build_spin_sop.F and are referred to as Tait-Bryan angles.

From wikipedia I see that such angles use a different convention compared to Euler.

What's the reference that demonstrates that it is possible to define spin rotations using the Tait-Bryan angles convention?

[andreamarini]

@muralidhar-nalabothula this is to follow the discussion we just had! Thanks so much!

[muralidhar-nalabothula]

The procedure is tricky as Sakurai discusses how the order of rotations must be carefully choosen (indeed he comments on the order reported in the Goldstein book on classical mechanics, stating that it is not consistent with QM).

This is incorrectly quoted. In Sakurai, a particular convention is used because, in this convention, we can write the rotation given in Eq. 3.85 (moving body axis) as a product of rotations around fixed Cartesian axes (x, y, z), as shown in Eq. 3.89. If we use Goldstein's convention, we cannot write it as a product of rotations around stationary axes. Since the whole point of Euler angles is to express any generic rotation as a product of rotation matrices for fixed Cartesian axes (we know the spin matrices for fixed axis rotations), using Goldstein's convention simply defeats the purpose.

Nevertheless, to prove that what Sakurai and YAMBO use are equivalent, consider the definition of a 2x2 unitary operator.

$$U(R) = e^{-i \mathbf{S} \cdot \hat{\mathbf{n}} \phi}$$ (Eq. 3.60 of sakurai)

This implies that the unitary matrix only depends on the axis of rotation and the angle. Since both conventions must lead to the same rotational matrix, the axis and angle do not change, and therefore, both should lead to the same 2x2 spin operator.

In the end, it is simply matter of taste what convention to use. Yambo uses a particular Tait–Bryan convention, where angles are defined about the three fixed axis.

Below are the angles used in sakurai (left), used in yambo (right). R is the 3x3 rotation matrix.

You can use the angle in the left column and use Eq.3.91 of sakurai and this should yield the same

[andreamarini]

@muralidhar-nalabothula Thanks! Let me go trough the math myself. I will be back to you ASAP.