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".
- How does the dynamics of $\langle x\rangle $ and $\langle p\rangle $ look like?
- How does the image change if the sign is changed on one of the coefficients?
- What happens if the angular frequency (here called "a") doubles or halves?
- What is the relationship between$\langle x\rangle $ and $\langle p\rangle $ from the graphs in Composer? How should they be connected theoretically?
- 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?
- 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?
- 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 !!!
- 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?
