A stationary state

Before looking at the superposition of two states, let us look at the time evolution of a single stationary state.

The initial state is thus taken proportional to an eigenstate j1 of energy E1.

y(x,t) = C j1(x)
The wavefunction is then shown in the left upper panel, the real part in green and the imaginary part in pink.

You can change the complex coefficient C either in the middle upper panel by changing ReC and ImC or in the right upper panel directly in the complex plane.

The lower panel shows the square of the wavefunction, |y(x,t)|2, i.e. the probability density of finding the particle at point x.

When you start the animation, by clicking on the PLAY button on the toolbar, you can observe the time evolution of the wavefunction:

y(x,t) = C exp(-iw1t) j 1(x)
where w1=2 p E1/h. This can be also written as
y(x,t) = C(t) j1(x)
with
C(t) = C(0) exp(-iw1t)
The coefficient C(t) therefore undergoes a circular motion in the complex plane, shown in the right upper panel. ReC(t) and ImC(t) follow a sinusoidal evolution, with a relative p/2 phase shift.

However, the square of the wavefunction does not change, because |C(t)|2 does not depend on time: The probability density does not change as a function of time. This is the reason such a state is called a stationary state.