In the book Introduction to Quantum Mechanics (by David Griffith) there is an Example 2.1:
Suppose a particle starts out in a linear combination of just two stationary states:
$$\Psi(x,0)~=~c_1\Psi_1(x)+c_2\Psi_2(x) $$
(Two keep it simple I´ll assume that the costants $c_n$ and the states $\Psi_n$ are real.) What is the wave function $\Psi(x,t)$ at subsequent times? Find the probability density, and describe its motion.
Solution: The first part is easy:
$$\Psi(x,t)~=~c_1\psi_1(x)e^{-iE_1/\hbar}\,+\,c_2\psi_2(x)e^{-iE_2/\hbar}$$
where $E_1$ and $E_2$ are the energies associated with $\psi_1$ and $\psi_2$. It follows that:
$$|\Psi(x,t)|^2~=~ (c_1\psi_1e^{iE_1/\hbar}+c_2\psi_2e^{iE_2/\hbar})(c_1\psi_1e^{-iE_1/\hbar}+c_2\psi_2e^{-iE_2/\hbar}) \\=~c_1^2\psi_1^2+c_2^2\psi_2^2+2c_1c_2\psi_1\psi_2cos[(E_2-E_1)t/\hbar]$$
Evidently the probability density oscillates sinusoidally. at an angular frequency $(E_2-E_1)/\hbar$; this is certainly not a stationary state. But notice that it took a linear combination of states (with different energies) to produce motion.
My problem is: The cosinus function equals zero at certain times. So, when I normalize the funkction at this time: $$\int_ {-\infty}^\infty c_1^2\psi_1^2+c_2^2\psi_2^2\,\text{d}x~=~1$$
On the other hand at certain times the cosinus function equals one. And then the integral $$\int_ {-\infty}^\infty c_1^2\psi_1^2+c_2^2\psi_2^2\,\text{d}x~+~\int_ {-\infty}^\infty2c_1c_2\psi_1\psi_2\,\text{d}x$$ should be bigger than 1. That seems impossible, so where is my mistake?
I am sorry when the question is unclear in some parts. I´m not used to speak english on science topics.
