Third update of aggregate notes for phy1520, Graduate Quantum Mechanics.

I’ve posted a third update of my aggregate notes for PHY1520H Graduate Quantum Mechanics, taught by Prof. Arun Paramekanti. In addition to what was noted previously, this contains lecture notes up to lecture 13, my solutions for the third problem set, and some additional worked practice problems.

Most of the content was posted individually in the following locations, but those original documents will not be maintained individually any further.

Determining the rotation angle and normal for a rotation through Euler angles

November 2, 2015 phy1520 , , ,

[1] pr. 3.9 poses the problem to determine the total rotation angle for a set of Euler rotations given by

\label{eqn:eulerAngleRotationAngleAndNormal:20}
\mathcal{D}^{1/2}(\alpha, \beta, \gamma)
=
\begin{bmatrix}
e^{-i(\alpha+\gamma)/2} \cos \frac{\beta}{2} & -e^{-i(\alpha-\gamma)/2} \sin \frac{\beta}{2} \\
e^{i(\alpha-\gamma)/2} \sin \frac{\beta}{2} & e^{i(\alpha+\gamma)/2} \cos \frac{\beta}{2}
\end{bmatrix}.

Compare this to the matrix for a rotation (again double sided) about a normal, given by

\label{eqn:eulerAngleRotationAngleAndNormal:40}
\mathcal{R}
= e^{-i \Bsigma \cdot \ncap \theta/2}
= \cos \frac{\theta}{2} I – i \Bsigma \cdot \ncap \sin \frac{\theta}{2}.

With $$\ncap = \lr{ \sin \Theta \cos\Phi, \sin \Theta \sin\Phi, \cos\Theta}$$, the normal direction in its Pauli basis is

\label{eqn:eulerAngleRotationAngleAndNormal:60}
\Bsigma \cdot \ncap
=
\begin{bmatrix}
\cos\Theta & \sin \Theta \cos\Phi – i \sin \Theta \sin\Phi \\
\sin \Theta \cos\Phi + i \sin \Theta \sin\Phi & -\cos\Theta
\end{bmatrix}
=
\begin{bmatrix}
\cos\Theta & \sin \Theta e^{-i \Phi} \\
\sin \Theta e^{i \Phi} & -\cos\Theta
\end{bmatrix},

so

\label{eqn:eulerAngleRotationAngleAndNormal:80}
\mathcal{R} =
\begin{bmatrix}
\cos \frac{\theta}{2} -i \sin \frac{\theta}{2} \cos\Theta & -i \sin \Theta e^{-i \Phi} \sin \frac{\theta}{2} \\
-i \sin \Theta e^{i \Phi} \sin \frac{\theta}{2} & \cos \frac{\theta}{2} +i \sin \frac{\theta}{2} \cos\Theta \\
\end{bmatrix}.

It’s not obvious how to put this into correspondence with the matrix for the Euler rotations. Doing so certainly doesn’t look fun. To solve this problem, let’s go the opposite direction, and put the matrix for the Euler rotations into the form of \ref{eqn:eulerAngleRotationAngleAndNormal:40}.

That is
\label{eqn:eulerAngleRotationAngleAndNormal:100}
\begin{aligned}
\mathcal{D}^{1/2}(\alpha, \beta, \gamma)
&=
\begin{bmatrix}
e^{-i(\alpha+\gamma)/2} \cos \frac{\beta}{2} & -e^{-i(\alpha-\gamma)/2} \sin \frac{\beta}{2} \\
e^{i(\alpha-\gamma)/2} \sin \frac{\beta}{2} & e^{i(\alpha+\gamma)/2} \cos \frac{\beta}{2}
\end{bmatrix} \\
&=
\begin{bmatrix}
\cos\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} & – \cos\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} \\
\cos\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} & \cos\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2}
\end{bmatrix} \\
i
\begin{bmatrix}
– \sin\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} & \sin\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} \\
\sin\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} & \sin\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2}
\end{bmatrix} \\
&=
\cos\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2}
+ i \sin\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} \sigma_x
– i \cos\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2} \sigma_y
– i \sin\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} \sigma_z
\end{aligned},

This gives us

\label{eqn:eulerAngleRotationAngleAndNormal:120}
\begin{aligned}
\cos\frac{\theta}{2} &= \cos\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} \\
\ncap \sin\frac{\theta}{2} &= \lr{ -\sin\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2}, \cos\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2}, \sin\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} }.
\end{aligned}

The angle is

\label{eqn:eulerAngleRotationAngleAndNormal:140}
\theta
= 2 \textrm{arctan} \frac{
\sqrt{\sin^2\frac{\beta}{2} + \sin^2\frac{\alpha+\gamma}{2} \cos^2\frac{\beta}{2}
}
}{\cos\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2}},

or
\label{eqn:eulerAngleRotationAngleAndNormal:180}
\boxed{
\theta = 2 \textrm{arctan} \frac{
\sqrt{\tan^2\frac{\beta}{2} + \sin^2\frac{\alpha+\gamma}{2}
}
}{\cos\frac{\alpha+\gamma}{2}
},
}

and the normal direction is
\label{eqn:eulerAngleRotationAngleAndNormal:160}
\boxed{
\ncap
=
\inv{\sqrt{1 – \cos^2\frac{\alpha+\gamma}{2} \cos^2\frac{\beta}{2} }}
\lr{ -\sin\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2}, \cos\frac{\alpha-\gamma}{2} \sin \frac{\beta}{2}, \sin\frac{\alpha+\gamma}{2} \cos \frac{\beta}{2} }.
}

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.