Exercise: Coherent States of the Quantum Oscillator

The applet models the quantum harmonic oscillator. The implied units are ½ℏω for energy, (ℏ÷mω)^½ for length, and 2÷ω for time, with ω the classical frequency of oscillation for a mass m. With these units the oscillator potential energy is simply V(x) = x², and the applet is capable of describing a wide range of physical phenomena by choosing m and ω appropriately. The applet also shows an initial matter wave describing the mass m bound by this potential. The initial wave is a gaussian function controlled by two parameters, a and d, that specify its width and location, respectively; the defaults a = ½, d = 0 are specified on the Math tab of the applet, and describe a matter wave initially centered in the oscillator well. In this exercise we explore the time evolution of this matter wave for different choices of the initial wave parameters.

1. Go to the Math tab, right-click anywhere in the formula field for ψ(x), then select "Animate..." from the popup menu. This brings up the Animator Editor. Switch back to the Graphics: [x] tab and re-position the editor as necessary to afford an unobstructed view of the waveform. Click the Play button at the top right of the editor to begin evolution. Complete control over the developing waveform is afforded by the remaining editor controls: Stop , Reverse , Restart , Step Ahead , and Step Back .
   
   
   Provision also exists for adjusting the refresh rate and the elapsed time between 'clock ticks'; both can be tweaked (with the 'clock' running) to achieve the desired visual effect.
   
   
2. Describe the evolving waveform. Note the color-for-phase plotting style (see 2nd Technical Note below), used here because the waveform at time t is generally complex-valued even though the initial wave is purely real. What do your observations imply for this, the case where the initial wave has a = ½ and d = 0?
3. Next move the initial wave to the right one unit. Begin by clicking the Animator Stop button to 'freeze' evolution (but leave open the editor), then switch back to the Math tab and click in the value field for the parameter d. Position the cursor as appropriate and type in the new value d = 1, then type Ctrl+Z to register the changed value. Now return to theGraphics: [x] tab. Note that the display is updated: to better compare with the d = 0 case, reset the 'clock' to zero by clicking the editor Restart button .
4. Continue the animation by clicking the editor Play button and again describe the evolving waveform. What is peculiar about this case? [For comparison, return to the Math tab and change the value of a to anything other than ½.] This is one of many so-called coherent states for the quantum oscillator. [Others can be generated by taking for the initial wavefunction any displaced stationary state of the quantum oscillator.]

Technical Notes

• The animation of time-dependent Schrödinger wavefunctions is accomplished using the Crank-Nicholson approximation to the system propagator, together with the Numerov method for the spatial integration. The boundary conditions enforced at the endpoints of the spatial interval depend on the problem at hand: for waves that are confined (as is the case here), the wave is required to vanish at the endpoints for all times (rigid-wall conditions); otherwise, transparent boundary conditions are employed to mimic an interval of infinite extent. Mathematical details and further references can be found in Am. J. Phys. 72(3), 351 (2004).
• A complex-valued function can be rendered as a single color-for-phase plot rather than two separate plots for the real and imaginary parts. In this display style, the phase of a complex number (the function value) is represented by color which, like phase angle, repeats with a definite period. [The 'colorwheel' follows the rainbow from red to green to violet, then back to red again through magenta.] In computer jargon, the phase is mapped to the hue (the amount of red, green, or blue) component of an hue-saturation-brightness color model.