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.
- How does evolution change if the sign is change or we add an imaginary coefficient ?
- 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?
- 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?
- 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?
- 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.
