Griffith's Quantum Mechanics example 2.1 states that:
Suppose a particle starts out in a linear combination of just two stationary states: $$ \Psi(x,0) = c_1 \psi_1(x) + c_2\psi_2(x) $$ Find the probability density and describe the motion.
The solution drives the probability density to be :
$ |\Psi(x,t)|^2 = 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]$
Here's my question:
Integrating that probability density for $ x= -\infty \ldots +\infty $ results in a time dependent value as opposed to a constant:
$$ 1 = \int_{-\infty}^{+\infty} |\Psi(x,t)|^2 dx=\int c_1^2\psi_1^2 \;dx + \int c_2^2\psi_2^2 \; dx+ 2c_1c_2\cos[(E_2 - E_1) t/\hbar]\int\psi_1\psi_2\;dx$$
Now all of the integrals are constants, so the end result is a function of time which makes the wave function impossible to normalize with a constant factor. So what is going on here? What am I not getting?
