Simple Harmonic Oscillator Exercises

[Screen shot of a SHO superposition state that oscillates back and forth in the well.]

Pre-set Demonstrations

You can access the pre-set initial states for the harmonic oscillator (SHO) via the textbox on the lower-left-hand side of the main simulation panel.  These show:

• SHO Two State: Loads: Loads a SHO two-state superposition (equal mix of ground state and first-excited state).
• SHO Squeezed: Loads: Loads a SHO coherent state with an initial average momentum.
• SHO Squeezed Wide: Loads a SHO coherent state with an initial average momentum.
• SHO Boosted Coherent: Loads a SHO coherent state with an initial average momentum.
• SHO Boosted Squeezed: Loads a SHO squeezed state with an initial average momentum.
• SHO Shifted Ground State: Loads the SHO ground state with an initial shift from x = 0.
• SHO Shifted Excited State: Loads the SHO first-excited state  with an initial shift from x = 0.

Exercises

1. What are the ground-state and first-excited-state energies?  Recall that we have scaled the problem such that ħ = m = 1 and chosen ω to be 4.

2. Since the time dependence of energy eigenstates is just, e^-iEnt/ħ, how long does it take the ground state and the first excited state to evolve in time back to their t = 0 values?

3. Select SHO Two State from the drop down menu.  Now set the dt to the ground state period divided by 10. The text field can accept simple mathematical operations such as /, *, pi, etc. Single step through the simulation and see if the wave function indeed has the same period as you calculated in Question 2.

4. Now select each of the SHO wave packets (SHO Squeezed, SHO Squeezed Wide, SHO Boosted Coherent, SHO Boosted Squeezed, SHO Shifted Ground State, SHO Shifted Ground State, SHO Shifted Excited State) from the drop down menu (you may have to scroll down in the drop down menu first before selecting a state). First look at all six wave packets.  Describe the similarities and differences between the packets' initial shapes. Now look at the SHO wave packets' time evolution. Describe what happens to each wave packet as it evolves in time.

5. Are there any special initial wave packets that behave differently than what you would have expected?   How do they behave?  Briefly describe what each of the wave packets time evolution looks like.  From your analysis, what is the difference between a coherent state and a squeezed state?

6. Are there any special wave packets that behave like a particle in a classical harmonic oscillator?

7. Compare and contrast the ISW wave packets' time evolution to that of the SHO wave packets' time evolution.  If you were given a wave packet and were not told which well the packet was in, what properties of the wave packet's evolution would suggest on well or the other?