How can I calculate the exponential of a non-diagonal matrix?

[maximus123]

Hello,

I have a problem where I'm given the following

[itex]H=-\frac{\hbar\Omega}{2}\sigma_x\quad\quad\quad\textrm{and}\quad\quad\quad\Psi(0)=\left|0\right\rangle\quad[/itex]
​

Where

[itex]\sigma_x=\begin{pmatrix}0 & 1\\1&0\end{pmatrix}\quad\quad\quad\textrm{and}\quad\quad\quad\left|0\right\rangle=\begin{pmatrix}1\\0\end{pmatrix}[/itex]
​

And in general

[itex]\Psi(t)=\textrm{exp}\left[-i\frac{H}{\hbar}t\right]\Psi(0)[/itex]
​

So

[itex]\Psi(t)=\textrm{exp}\left[i\frac{\Omega t}{2}\sigma_x\right]\left|0\right\rangle[/itex]
​

The problem is I need to get from here to

[itex]\Psi(t)=\textrm{exp}\left[i\frac{\Omega t}{2}\sigma_x\right]\left|0\right\rangle=\begin{pmatrix}\textrm{cos}(\Omega t/2)\,\,\,&i\textrm{sin}(\Omega t/2)\\i\textrm{sin}(\Omega t/2)\,\,\,&\textrm{cos}(\Omega t/2)\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}\\\\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,=\begin{pmatrix}cos(\Omega t/2)\\isin(\Omega t/2)\end{pmatrix}
[/itex]
​

I can't work out how to get to this cos and sine matrix. I tried this

[itex]\Psi(t)=\textrm{exp}\left[i\frac{\Omega t}{2}\sigma_x\right]\left|0\right\rangle=\left\lbrace\textrm{cos}\left(\frac{\Omega t}{2}\sigma_x\right)+i\textrm{sin}\left(\frac{\Omega t}{2}\sigma_x\right)\right\rbrace\left|0\right\rangle\\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad=\left\lbrace\textrm{cos}\begin{pmatrix}0 & \frac{\Omega t}{2}\\\frac{\Omega t}{2} & 0\end{pmatrix}+i\textrm{sin}\begin{pmatrix}0 & \frac{\Omega t}{2}\\\frac{\Omega t}{2} & 0\end{pmatrix}\right\rbrace\begin{pmatrix}1\\0\end{pmatrix}[/itex]
​

Beyond this I cannot see how to get from here to

[itex]\begin{pmatrix}\textrm{cos}(\Omega t/2)\,\,\,&i\textrm{sin}(\Omega t/2)\\i\textrm{sin}(\Omega t/2)\,\,\,&\textrm{cos}(\Omega t/2)\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}[/itex]
​

Any help would be really appreciated

 

[DrClaude]

There might be more than one way to approach the problem, but I would proceed thus. You can calculate the exponential of a matrix easily if the matrix is diagonal. For
$$
A = \left( \begin{array}{cc} a_1 & 0 \\ 0 & a_2 \end{array} \right)
$$
$$
\exp(A) = \left( \begin{array}{cc} e^{a_1} & 0 \\ 0 & e^{a_2} \end{array} \right)
$$
Can you find the rotation that brings you from the z-basis to the x-basis?

 

[maximus123]

Thanks for responding,

So based on what you've said for the exponential of a matrix the expression in my original post would become

[itex]
\Psi(t)=\textrm{exp}\left[i\frac{\Omega t}{2}\sigma_x\right]\left|0\right\rangle=\textrm{exp}\left[\begin{pmatrix}0 & i\frac{\Omega t}{2}\\i\frac{\Omega t}{2} & 0\end{pmatrix}\right]\left|0\right\rangle=\begin{pmatrix}0 & \textrm{exp}\left[i\frac{\Omega t}{2}\right]\\\textrm{exp}\left[i\frac{\Omega t}{2}\right] & 0\end{pmatrix}\left|0\right\rangle\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\begin{pmatrix}0 & \textrm{cos}\left(\frac{\Omega t}{2}\right)+i\textrm{sin}\left(\frac{\Omega t}{2}\right)\\\textrm{cos}\left(\frac{\Omega t}{2}\right)+i\textrm{sin}\left(\frac{\Omega t}{2}\right) & 0\end{pmatrix}\left|0\right\rangle

[/itex]

is that correct? It doesn't seem much closer to the intended end result.

When you asked about the rotation are you referring to the relationships [itex]\sigma_y\sigma_z=-i\sigma_x[/itex] etc.?

 

[DrClaude]

Calculating the exponential of a matrix by taking the exponential of the elements only works for a diagonal matrix. You can show this by considering the series expansion of the exponential function.

You do not have a diagonal matrix, this is why I said you need to look at the rotation matrix that will bring you from the z basis to the x basis, in which ##\sigma_x## is diagonal.