First Order Correction to wave function in ground state I am looking at a spin 1/2 particle in a magnetic field.  This has Hamiltonian
$$H=-\mu s\cdot B_0$$
For simplicity, assume $B_0=B_0\hat z$ so $H=-\mu B_0$.  I then apply a perturbative magnetic field such that 
$$V'=-\mu B_1 s_x$$
First I wanted to compute $E^{(1)}$
$$E^{(1)}_n=\langle\psi_n^{(0)}|-\mu B_1s_x|\psi_n^{(0)}\rangle=\mp \mu B_1 \hbar/2$$
Now I am looking to find the first order correction to the ground state wavefunction.  I know that this is given as
$$\psi^{(1)}_n=\sum_{n\neq n'} \psi^{(0)}_{n'}\frac{\langle\psi_{n'}^{(0)}|-\mu B_1s_x|\psi_{n}^{(0)}\rangle}{E_n^{(0)}-E_{n'}^{(0)}}$$
I am confused as to how to treat the summation.  The only term I would get is if $n=n'$, but that would be degerate.  So I am thinking that this first order correction is 0.  Is this correct?
 A: Spin1/2 particle
Ususally, in this kind of Hamiltonian, people uses $s=s_z$, where
$$s=s_z=\left[ \begin{array}{cc}
1 & 0 \\
0 & -1\end{array} \right].$$
Then, your unperturbed hamiltonian $H_0$ is: $$H_0=-\mu s\cdot B_0 = -\mu  \left[ \begin{array}{cc}
1 & 0 \\
0 & -1\end{array} \right]B_{0,z}. $$
Then the eigen vectors of energy are:
$$|\psi^0_+\rangle=\left[ \begin{array}{c} 1 \\ 0\end{array} \right],$$ 
$$|\psi^0_-\rangle=\left[ \begin{array}{c} 0\\ 1  \end{array} \right].$$ 
Perturbation solution
Then you want to compute $|\psi_+\rangle$ and  $|\psi_-\rangle$ for the perturbed Hamiltonian $H=H_0-\mu B_1 s_x$, where $$s_x=\left[ \begin{array}{cc}
0 & 1 \\
1 & 0\end{array} \right].$$
As you said, you have to compute the following quantities (note I use $+,-$ instead of $n=0,1$. Which became:
$$\psi^{(1)}_+=\sum_{n\neq +} \psi^{(0)}_{n'}\frac{\langle\psi_{n'}^{(0)}|-\mu B_1s_x|\psi_{+}^{(0)}\rangle}{E_+^{(0)}-E_{n'}^{(0)}}=\psi^{(0)}_{-}\frac{\langle\psi_{-}^{(0)}|-\mu B_1s_x|\psi_{+}^{(0)}\rangle}{E_+^{(0)}-E_{-}^{(0)}}$$
$$\psi^{(1)}_-=\sum_{n\neq -} \psi^{(0)}_{n'}\frac{\langle\psi_{n'}^{(0)}|-\mu B_1s_x|\psi_{-}^{(0)}\rangle}{E_-^{(0)}-E_{n'}^{(0)}}=\psi^{(0)}_{+}\frac{\langle\psi_{+}^{(0)}|-\mu B_1s_x|\psi_{-}^{(0)}\rangle}{E_-^{(0)}-E_{+}^{(0)}}$$
Put here vectors and matrices we just found and let me know if you get zero.
