#### 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
j_{1} of energy E_{1}.

y(x,t) = C j_{1}(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 *Re*C and *Im*C 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(-iw_{1}t) j
_{1}(x) where w_{1}=2
p E_{1}/h. This can be also written as
y(x,t) = C(t) j_{1}(x)
with C(t) = C(0) exp(-iw_{1}t)
The coefficient C(t) therefore undergoes a circular motion in the complex
plane, shown in the right upper panel. *Re*C(t) and *Im*C(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.