Eigenstates can also be calculated by "imaginary time" propagation by considering the equation:
At long times,
since all higher 
have decayed away to a much greater extent than 
The method is called "imaginary time propagation" since
corresponds to t in the ordinary form of the propagator. The method is also called the "relaxation" method, since it, is based on the faster decay of the higher levels. Here the method is used recursively: after the lowest energy eigenstate,
is calculated, the component of this state in the initial wavepacket is removed. After another imaginary time propagation, only
survives. This process is repeated here for the first 9 eigenstates of the harmonic oscillator.
At long times,
since all higher
have decayed away to a much greater extent than The method is called "imaginary time propagation" since
corresponds to t in the ordinary form of the propagator. The method is also called the "relaxation" method, since it, is based on the faster decay of the higher levels. Here the method is used recursively: after the lowest energy eigenstate,
is calculated, the component of this state in the initial wavepacket is removed. After another imaginary time propagation, only
survives. This process is repeated here for the first 9 eigenstates of the harmonic oscillator.