Understanding Dirac notation from matrix representation

Question

I want to solve for the state after qubit rotation by operator, $U_\alpha = e^{i\alpha\sigma_1}$ = $\begin{bmatrix} \cos\alpha& i\sin\alpha\\i\sin\alpha & \cos\alpha\end{bmatrix}$, if qubit is initially in state $\left|0\right\rangle$

Using matrix representation, I can easily solve this to get;

$ U_\alpha \left|0\right\rangle = \begin{bmatrix} \cos\alpha& i\sin\alpha\\i\sin\alpha & \cos\alpha\end{bmatrix} \begin{bmatrix} 1\\0 \end{bmatrix} = \begin{bmatrix} \cos\alpha\\i\sin\alpha \end{bmatrix} = \cos\alpha\left|0\right\rangle + i\sin\alpha \left|1\right\rangle $

I have just started learning the Dirac notations and cannot seem to understand how to solve this using bra and ket.

How can I find the same using Dirac notations?

Answer 1

Given that $O_{ij}$ is the entry of the matrix $O$ at the $i$-th row and $j$-th column, the operator $\hat{O}$, from which you obtain the matrix $O$ by projecting it on elements $|i\rangle$ and $|j\rangle$ out of an orthonormal basis spanning the Hilbert space on which $\hat{O}$ acts, can be written as

$$\hat{O}=\sum_{ij}\langle i |\hat{O}|j\rangle |i\rangle \langle j|=\sum_{ij}O_{ij} |i\rangle \langle j|.$$

So, in your case $$\hat{U}_{\alpha}=\cos(\alpha)|0\rangle \langle 0|+ i\sin(\alpha)|0\rangle \langle 1|+i\sin(\alpha)|1\rangle \langle 0|+ \cos(\alpha)|1\rangle \langle 1|$$ acts on $|0\rangle$ as $$\hat{U}_{\alpha}|0\rangle=\cos(\alpha)|0\rangle \langle 0|0\rangle+ i\sin(\alpha)|0\rangle \langle 1|0\rangle +i\sin(\alpha)|1\rangle \langle 0|0\rangle + \cos(\alpha)|1\rangle \langle 1|0\rangle.$$ Since $\langle 0|0\rangle=1$ and $\langle 1|0\rangle=0$, $$\hat{U}_{\alpha}|0\rangle=\cos(\alpha)|0\rangle + i\sin(\alpha)|1\rangle.$$