Exercise: The Quantum Bouncer

The applet models a mass m (the bouncer) in a uniform gravitational field directed along the x-axis, and described by the potential energy V(x) = mgx for x > 0. A virtually impenetrable surface at x = 0 marks the position of the 'floor'. The choice of Flex Units here means the scales for distance (d) and energy (E) can be adapted to the application at hand; here they allow for the incredibly small scales of energy and distance needed to describe the low-lying states of a macroscopic bouncer, like a tennis ball. In these units, distance and energy scales are related as E·d² = ℏ²÷2m. We take d = 10⁻²² meters [m] to give an energy unit E ~ 10⁻²³ Joules [J] for a mass m = 50 grams. The gravity force on the bouncer F = mg is the slope of the linear segment of the potential curve, about 0.5 J/m, or simply 5 in Flex units. This value for F appears on the Math tab of the applet, along with U, the minimum energy the bouncer needs to penetrate the floor. The default U = 10000 (~10⁻¹⁹ J) is unrealistically small, but still large enough to accurately model the low-lying states of the bouncer. On the tab labeled Graphics: [x] the bouncer potential energy function V(x) is plotted over the interval [–1, +10]. The listing to the right of the graph includes placeholders for three stationary states of the bouncer. In this exercise we will find the three lowest stationary states of the bouncer and confirm that energy is quantized even for macroscopic objects like a bouncing tennis ball.

1. Show the first waveform in the list (labeled ψ₀) by right-clicking its placeholder, clicking on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the OK button. The displayed wavefunction has a noticeable discontinuity. Since quantum wavefunctions must be everywhere continuous, the energy of this state cannot be one of the allowed energies for the bouncer.
2. Adjust the energy of this state upward from zero to eliminate the discontinuity. Go to the Math tab, where the energy of this state is recorded as E₀ = 0. Right-click anywhere in the value field for E₀ and choose "Edit Parameter..." from the popup menu to bring up the Energy Editor. Return to the Graphics: [x] tab and re-position the editor so as to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the energy field (labeled E) while observing the waveform. Your goal is to reduce the discontinuity to an imperceptible level. 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the energy field. The actual wave mismatch, expressed as a fraction of the wave value, is recorded in the editor tolerance field, labeled δψ. When no discontinuity is evident, the energy is 'allowed' and the wavefunction is one of the stationary states for the bouncer. The reset button () at the lower right changes the point of discontinuity, and should be used when nearing a correct energy – see Technical Notes below. Count the number of nodes for the wavefunction to see which stationary state you have found. We seek the ground state (nodeless), first excited state (one node), or the second excited state (two nodes). If you have found another, continue searching at lower energies. Finish by selecting OK to end the edit session with the current settings.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   
3. Repeat the above procedure for the second and third placeholders in the list, using them to find the remaining stationary states, i.e., those not found in the preceding step. Recall that the stationary state energies you have found are in units of 10⁻²³ J!
4. For each stationary state, there is a most probable height from the floor for the bouncer, identified as that value of x for which the probability amplitude |ψ(x)| is largest. Locate this position for each state by right-clicking anywhere on the graph, selecting "Trace On/Off" from the popup menu, and surveying the results. Remember that all numerical values for x are in units of 10⁻²² m! Compare the quantum predictions with their classical counterparts, discussing any discrepancies in the context of Bohr's Correspondence Principle. [Classically, the bouncer is most likely to be found at the top of its flight, where the speed drops to zero.]

Final form of applet...

Technical Notes

• The Numerov method is used to construct stationary waves for a given (trial) energy. Using boundary conditions derived from the correct asymptotic form first at the left endpoint, and then again at the right endpoint, the Schrödinger equation is integrated inward to a preset match point. The right-side wave is then scaled to make its slope agree with that of the left-side wave at the match point. If the trial energy is allowed, the wavefunction values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the editor tolerance field, labeled δψ. The technique is most accurate when the match point coincides with an extremum of the wavefunction. Each press of the reset button re-positions the match point to an extremum of the current waveform.