Suppose a particle starts out in a linear combination of just two stationary states:
\Psi \left(x,0\right) = c_{1} \psi _{1}\left(x\right) + c_{2} \psi _{2}\left(x\right) .
(To keep things simple I’ll assume that the constants c_{n} and the states \psi _{n}\left(x\right) are real.)What is the wave function \Psi \left(x,t\right) at subsequent times?Find the probability density and describe its motion.
The first part is easy:
\Psi \left(x,t\right) = c_{1} \psi _{1}\left(x\right) e^{{-iE_{1} t}/{h}} + c_{2} \psi _{2}\left(x\right) e^{{-iE_{2} t}/{h}} ,
where E_{1} and E_{2 } are the energies associated with \psi _{1} and \psi _{2} . It follows that
\left|\Psi(x,t) \right|^{2} = \left( c_{1} \psi _{1} e^{{-iE_{1} t}/{h}} + c_{2} \psi _{2} e^{{-iE_{2} t}/{h}} \right) \left( c_{1} \psi _{1} e^{{-iE_{1} t}/{h}} + c_{2} \psi _{2} e^{{-iE_{2} t}/{h}} \right)= c_{1} ^{2} \psi_{1}^{2} + c_{2} ^{2} \psi_{2}^{2} + 2 c_{1} c_{2} \psi_{1} \psi_{2} \cos \left[{\left(E_{2}-E_{1} \right) t}/{h}\right] .
The probability density oscillates sinusoidally, at an angular frequency \omega = {(E_{2}-E_{1} )}/{h} ; this is certainly not a stationary state. But notice that it took a linear combination of stationary states (with different energies) to produce motion.