- #1
Danny Boy
- 49
- 3
What I am interested in doing, is considering the angular momentum eigenstate for a spin ##1## system: ##|J=1, M=1\rangle = \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}##, forming the coherent state ##|CS \rangle = \begin{bmatrix}
0.5 \\
-\frac{i}{\sqrt{2}} \\
-0.5
\end{bmatrix}## by taking the rotation ##\text{exp}(-\frac{i \pi}{2}\hat{J}_x)\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} = \begin{bmatrix}
0.5 \\
-\frac{i}{\sqrt{2}} \\
-0.5
\end{bmatrix}##.
I then want to work out the modulus squared of the overlap of the coherent state with a rotation of the coherent state about the ##x##-axis and the ##z##-axis for various values of ##\theta## and ##\phi##: $$|\langle CS| \text{exp}(-i \phi \hat{J}_z) \text{exp}(-i \theta \hat{J}_x)|CS\rangle|^2.$$ The result I get is http://farm5.staticflickr.com/4211/34692246053_e0fd2d7cd8_b.jpg. Is this the type of plot you would have predicted? I would have expected something more circular symmetric?
Thanks for any assistance.
1 \\
0 \\
0
\end{bmatrix}##, forming the coherent state ##|CS \rangle = \begin{bmatrix}
0.5 \\
-\frac{i}{\sqrt{2}} \\
-0.5
\end{bmatrix}## by taking the rotation ##\text{exp}(-\frac{i \pi}{2}\hat{J}_x)\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} = \begin{bmatrix}
0.5 \\
-\frac{i}{\sqrt{2}} \\
-0.5
\end{bmatrix}##.
I then want to work out the modulus squared of the overlap of the coherent state with a rotation of the coherent state about the ##x##-axis and the ##z##-axis for various values of ##\theta## and ##\phi##: $$|\langle CS| \text{exp}(-i \phi \hat{J}_z) \text{exp}(-i \theta \hat{J}_x)|CS\rangle|^2.$$ The result I get is http://farm5.staticflickr.com/4211/34692246053_e0fd2d7cd8_b.jpg. Is this the type of plot you would have predicted? I would have expected something more circular symmetric?
Thanks for any assistance.