omega = 5; beta = 2;
sol = NDSolve[{
-1/2*D[psi1[x, t], {x, 2}] + omega^2 Cos[\[Theta]] psi1[x, t] +
beta psi2[x, t] == I D[psi1[x, t], t],
-1/2*D[psi2[x, t], {x, 2}] - omega^2 Cos[\[Theta]] psi2[x, t] +
beta psi1[x, t] == I D[psi2[x, t], t],
psi1[0, t] == psi1[2 Pi, t],
Derivative[1, 0][psi1][0, t] == Derivative[1, 0][psi1][2 Pi, t],
psi2[0, t] == psi2[2 Pi, t],
Derivative[1, 0][psi2][0, t] == Derivative[1, 0][psi2][2 Pi, t],
psi1[x, 0] == E^(-omega/2* (-(Pi/2) + x)^2),