Mean values timeevolution

Exercise 4

In this exercises you will investigate
Open the file "Exercise 4 - mean values timeevolution.flow". Then a node diagram appears, much like it from exercise 3. In addition to a "Position Plot" which shows the wave function for the current time, there are also calculations of the mean values $\langle x\rangle $and $\langle p\rangle $.
Start the time development by clicking on the green play button at the top left. Note the inital wave function, which is defined as a linear combination of stationary states in the node "Linear combination".

  1. How does the dynamics of $\langle x\rangle $ and $\langle p\rangle $ look like?
  2. How does the image change if the sign is changed on one of the coefficients?
  3. What happens if the angular frequency (here called "a") doubles or halves?
  4. What is the relationship between$\langle x\rangle $ and $\langle p\rangle $ from the graphs in Composer? How should they be connected theoretically?
  5. Try to change the potential of eg.0.5 * a ${}^\wedge$ 2 * x ${}^\wedge$ 4 and repeat the above question. Can you get more "wildness" into the time development for $\langle x\rangle $and $\langle p\rangle $? If so, why? What is special about the harmonic oscillator?
  6. To shine a little more light on the above quirks of the harmonic oscillator, consider how $\langle x\rangle $ behaves as a function of time for an arbitrary linear combination of stationary states. Which modes / crossover plays a role in the sandwich formula when the x or p operator is expressed through the raising and lowering operators?
  7. Try the following start conditions for the harmonic oscillator (remember to return the potential to 0.5 * a ${}^\wedge$ 2 * x ${}^\wedge$ 2): $c_0 = 0.74, c_1 = 0.60, c_2 = 0.01, c_3 = -0.27, c_4 = -0.16$. Get "Position Plot" to show $\langle x\rangle \pm \sigma_x$ as well. What happens to the spread as a function of time?
    The condition is called "amplitude-squeezed", since $\sigma_x$ is small when the amplitude $\langle x\rangle $ is large. On the other hand, one must live with a large $\sigma_x$ when $\langle x\rangle $ is small !!!
  8. Plot $\sigma_x$,$\sigma_p$ and even their product to check whether Heisenberg's uncertainty relationship is met. What are the values of these spreads and their product for the ground state?