Rotation matrix with Dirac notation Considering I've got an orthonormal base, composed of three states $ |i\rangle,|j\rangle$ and $|k\rangle$, I've got a question where it asks me to determine the rotation matrix, for a rotation by an angle $\theta$ around the last state, $|k\rangle$. However, it doesn't give me any information on what these states look like. How could I do it?
 A: rotation is inside a plane. If you have $3$ axes, then saying in which plane you rotate is identical to saying about which axis you rotate, with the plane being the one perpendicular to the axis. For example in standard Euclidean $3$-space, using Cartesian coordinate $x$, $y$ and $z$, you can either say that you rotate in the $x-y$ plane or rotate about the $z$ axis. In dimensions higher than three you cannot define a rotation about a single axis, as all axes have many (i.e. infinite) planes perpendicular to them.
But you are in three-dimensional space, so this is good! Now we can note that a rotation about an axis is characterized by that it leaves this axis unchanged. So a rotation about the $z$ axis will leave all $z$-components of a vector as they are.
Similarly for your case, then, a rotation about the (axis spanned by the) $|k\rangle$ state, will leave it intact
$$ R_{|k\rangle}(\theta) |k\rangle = |k\rangle$$
where we use $R_{|k\rangle}(\theta)$ to note a rotation about that axis by an angle $\theta$.
The next step is more open to interpretation, because without clock-wise orientation, we can define a rotation by degree $\theta$ in two opposite ways. But going back to rotations about the $z$-axis, we know that when we rotate about that axis, i.e. in the $x-y$ plane, by an angle $\theta$, the $x$ and $y$ components change as
$$ R_z(\theta) \hat{x} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}$$
$$ R_z(\theta) \hat{y} = \cos(\theta) \hat{y} - \sin(\theta) \hat{x}$$
note that we chose here a direction of rotation (the opposite direction can be inferred by taking $\theta \to -\theta$). So by analogy we can say
$$ R_{|k\rangle}(\theta) |i\rangle = \cos(\theta) |i\rangle + \sin(\theta) |j\rangle$$
$$ R_{|k\rangle}(\theta) |j\rangle = \cos(\theta) |j\rangle - \sin(\theta) |i\rangle$$
with the states spanning their respective axes. Now all you need to do is sort it out into a $3\times 3$ matrix.
