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Section 1.2: Animations, Units, and Measurement

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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.

Please wait for the animation to completely load.

Although computer simulations allow precise control of parameters, their spatial resolution is not infinite. Whenever these data are presented on screen as numeric values, they are correct to within the last digit shown. Start Animation 4 by clicking the "set the state: t = 0" button and follow the procedures below to make position measurements. What is shown in the animation is the time development of the same states shown in the previous animations (ħ = 2m = 1).

Start by click-dragging the mouse inside the animation to make measurements. Try it. Place the cursor in the energy eigenfunction graph and hold down the left mouse button. Now drag the mouse around to see the x and y coordinates of the mouse change in the lower left-hand corner of the animation. Notice the way the coordinates change. In addition, these measurements cannot be more accurate than one screen pixel.  This means that depending on how you measure the position of an object you may get a slightly different answer than another student in your class.

In this animation we have also given you two choices for the time scale. In the default animation, we have not changed the time scale and the ground state takes 2/π = 0.6366 to go back to its t = 0 position. When you click in the check box and click the "set the state: t = 0" button, you get the same animation, but with the energy, and therefore the time, scaled. We often do this to make time measurements easier to understand. In this example we have scaled the time so that the ground state returns to its t = 0 position when t = 1.  This time is the so-called revival time for the infinite square well.2

1 It is also common to see the convention that ħ = m = e2 = 1, which are called atomic units and is due to Hartree. In atomic units, the distance scale is given in Bohr radii (a0 = ħ2/me2 = 1) and energy is given in Hartrees (1 Hartree = 2 Rydbergs = me4/ħ2 = 1).   We choose ħ = 2m = 1 because the combination ħ2/2m occurs in the Schrödinger equation.
2 This is discussed in Chapter 10.

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