H133: 1094 Session 4

Write your name and answers on this sheet and hand it in at the end.
After the indicated time, move on to the next activity, even if you are not finished!

1. Quanton in a Box [12 min.]

Start up the PhET applet "Quantum Bound States". (Start->Programs->PhET, choose "Quantum Phenomena" from the left menu, and click on the Quantum Bound States icon). This applet shows energy levels and the corresponding wavefunctions and probability densities for the energy eigenvectors of different potentials. The "square well" potential that comes up initially is a finite-depth version of the "quanton in a box" potential.

1. What force would a Newtonian (i.e., classical) particle feel at each x position? Note that the wavefunctions don't go to zero at the "walls" of the well; could this happen classically? Explain.
   
   
   
   
2. For what energies would a quanton be "unbound" (i.e., not confined to the well)?
   
   
   
3. Roughly predict the energy of the n=2 state (second-lowest energy) by estimating its wavelength from the simulation and using this to find the momentum and then the kinetic energy. (To show the wave function, click on the 2nd level, switch the Display to "Wave Function", and press "Pause" to stop time.) Compare to the value given by the simulation.
   
   
   
   
4. Predict (and explain) what will happen to the energy levels if you make the box shallower. [Hint: Does the wave function penetrate the walls more or less?] Check your prediction by selecting "Configure Potential" and reducing the "Height". Try a new explanation if you were wrong the first time.
   
   
   
   
5. Do two-minute problem Q8T.3. [See Equation (Q8.5b) for reference.]
   
   
   
   

2. Harmonic Oscillator [12 minutes]

Change the Potential Well using the pulldown menu on the upper right to Harmonic Oscillator. This has the potential energy k*x²/2, which is the same as a spring.

1. For what energies would a quanton be "unbound" (i.e., not confined to the harmonic oscillator)?
   
   
2. Vibrational levels of molecules are described by a harmonic oscillator potential. Do problem Q8B.4.
   
   
   
   
3. Time dependence I. Based on Rule 6, what is the time dependence of the probability density for the state with E2? (Find the time dependence of the wave function psi2(x,t) for a state composed only of eigenvector |E2> and then calculate the probability density.) Does this agree with the simulation?
   
4. Time dependence II. Now use "Superposition State" to make a state with equal parts E0 and E2 (so c0 and c2 should be equal). Choose "Normalize" and then "Apply". Measure the period of the probability density using the clock in the lower left. Bonus: How is the result related to the energies E0 and E2?
   
   
   
   

3. Models of the Hydrogen Atom [10 minutes]

Start up the PhET applet "Models of the Hydrogen Atom". This applet simulates an experiment in which you shoot photons of many different wavelengths ("White") or a single wavelength ("Monochromatic") at a box with hydrogen gas. With the switch on "Experiment", you can see what actually happens outside the box. With the switch on "Prediction", you can compare various models ranging from more classical to more quantum mechanical to see what they predict.

1. Turn on the electron gun and "Show Spectrometer". Take a look first at "Experiment", then switch to "Prediction". Try each of the models. Note that some are in clear contradiction to the experiment (e.g., the Billiard Ball model predicts many photons bouncing backwards). For the last three, "Show electron energy diagram".
2. For the "Classical Solar System" model, there is quickly a "kaboom". What is happening and why? (Click on the icon again to replay it.) How is this avoided in quantum mechanical models?
   
   
   
3. How does the deBroglie model compare to the Bohr model? (E.g., similarities and differences.)
   
   
   
   
4. Which of the models will (approximately) reproduce the known hydrogen spectrum?
   
   
   

4. Discharge Lamps [12 minutes]

Start up the PhET applet "Neon Lights and Other Discharge Lamps". This simulates the type of light we saw in class. You can show "One Atom" of gas or "Multiple Atoms". Start with "One Atom".

1. Switch from "Single" Electron Production to "Continuous" and click on the Spectrometer. What is the source of energy to move an atom from the ground state to an excited state?
   
   
   
   
2. The Spectrometer shows only one wavelength of light. Why? What do you need to change to get more wavelengths? (Try it!) What if the energy of collision is less than the n=2 level (marked with a circled number 1)?
   
   
   
   
3. Switch to "Multiple Atoms" and click on the Spectrometer again. Identify the wavelength of the red line and show that the Bohr model quantitatively predicts it. [Hint: see Q8X.5 on pg.156.]
   
   
   
   
4. Take a look at other atoms besides hydrogen. Why do they have more visible lines in their spectra?