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# Section 1.2: Animations, Units, and Measurement

Most physics problems are idealizations of actual physical situations. These idealizations can take many forms in quantum mechanics. One way we can simplify a problem is by the choice of convenient units. This is especially important in quantum physics and in the animations we show in Physlet Quantum Physics. Restart.

Consider Animation 1 in which we have an electron (me = 9.109 × 10−31 kg) confined to a one-dimensional box with a length of the Bohr radius (a0 = 5.3 × 10−11 m).  According to quantum mechanics (see Chapter 10), the energy spectrum for such a situation is quantized such that

En = n2π2ħ2/2mL2 (1.1)

where ħ = 1.055 x 10−34 J·s is Planck's constant divided by 2π, L is the length of the box (here one Bohr radius), and n is a positive integer. For the current situation, the energy spectrum is En = n2 (2.146 × 10−17) J, which is depicted in Animation 1.   You can click-drag in the energy spectrum on the left of the animation to change the energy level (only the first 10 are shown) and as you do so, the current energy level turns from green to red. What do you notice about the energy spectrum?   You should notice that the numerical value for the energy is not very helpful in determining the functional form of the energy levels if you did not know it already. Can we make the functional form of the energy spectrum more transparent?

We can use units that make the physics more transparent by scaling the problem accordingly. This entails setting certain variables in the problem to simpler values (such as 1) and working in a dimensionless representation. In Animation 2 we have used one such common choice of units used in numerical simulations: ħ = 2m = L = 1.1  In these units, the length scale is a0/2 (0.265 × 10−10 m) and the energy scale is 4 Rydbergs or 2 Hartrees (54.4 eV).  Notice that the energy spectrum is now somewhat simpler: at the very least we do not have a factor of 10−17 in the energy anymore. We can also use other scaling conditions to simplify the energy spectrum even further as shown in Animation 3.  Here we set the combination π2ħ2/2mL2 = 1 which now scales the energy spectrum in units of the ground-state energy. What can you say about the energy spectrum now?  By choosing appropriate units, it has become clear that the energies are an integer squared times the ground-state energy.

In general, you should look for the units specified in the problem (whether from your text or from Physlet Quantum Physics 2E on ComPADRE): all units are in boldface in the statement of the exercise.

Pick an n = ; then

Check, then click the "set the state" button to see the alternate time scale.