# Atom state vectors kets

An atom with two energy levels has 2 states (excited and ground), represented by kets $|e\rangle$ and $|g\rangle$ respectively. The atom has energy $\frac{1}{2}E_\theta$ when excited and $-\frac{1}{2}E_\theta$ when in ground state.

Suppose it is prepared in the state $$|\Psi\rangle = \frac{1}{5}((2-i)|e\rangle + (4+2i)|g\rangle)$$

How do I show that the physical state of the system can be equally well described by the state vector written in the form

$$|\Psi\rangle = \cos(\frac{1}{2}\theta)|e\rangle + \sin(\frac{1}{2}\theta)e^{i\phi}|g\rangle$$

i.e determine the values of $\theta$ and $\phi$

Find the value of $\frac{1}{5}(2-i)$ in $r_{1}$$e^{i\alpha} and \frac{1}{5}(4+2i) in r_{2}$$e^{i\beta}$ form.
Now write $\frac{1}{5}(2-i)$=$\cos(\frac{1}{2}\theta)e^{i\alpha}$ and
$\frac{1}{5}(4+2i)$=$\sin(\frac{1}{2}\theta)e^{i\beta}$
so $\theta = 2Cos^{-1}(r_{1})$ and $\phi=\beta -\alpha$