Probability density timeevolution

Exercise 3

In this exercise you will investigate the time evolution of superposition in a harmonic potential. 
Open the file "Exercise 3 - probability density timeevolution.flow". Then a node diagram appears with many of the same parts as in the previous exercises. There is also a field called "Time evolution", which is just a "pre-loop". Inside this you see the following:

  • Time Evolution: The wave function is time-developed using the time-dependent Schr\"{o}dinger equation.
  • Position Plot: The current wave function is displayed.

Start the time development by clicking on the green play button at the top left. Observe the development of the wavefunction.

  1. How does evolution change if the sign is change or we add an imaginary coefficient ?
  2. Maintain the wave function as an equal linear combination of the two lower energy eigenvalues. If you change the angular frequency of the potential (here called 'a')what happens to; the potential wave function? the magnitude of the fluctuation of $\langle x\rangle $? and the time dynamics?
    Explain your observations?
  3. Try to include more states in your linear combination. Can you get $\langle x\rangle$ to be static even if the wave function develops in time?
  4. Try to include eg. the 5 lowest eigenstates $c_n$, with $n = 0,1,2,3,4$. Select the coefficients as $c_n = \lambda n / \sqrt {n!}$, Where $\lambda = 0.5$ is a suitable value. Is there anything special about the resulting wave function?
    Note, composer itself ensures that the coefficients are normalized - the above $| c_n |^2$ corresponds (after correct normalization) to a Poisson distribution, and the wave function is in practice what is called a coherent state. 
    What happens if $\lambda$ doubles?
  5. You can also get Composer to calculate the integral over $| \psi (x, t) |^2$. Enter the file "Exercise 3 - probability integral timeevolution.flow" and run the program. Try to include more coefficients in the linear combinations and see if you have a good intuition about the dynamics.