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Lecture 16: Quantized energies and spectra
March 27, 2025
Reading Assignment
- Read: Supplementary Reading Ch 4
Objectives
- (Continuing objective) Describe applications of the concepts of quantum mechanics to everyday “real-life” situations.
- For a particle in an infinite square well potential, draw the allowed wavefunctions \(\psi_n\) and express them mathematically. From the wavefunction, use Schrödinger's equation to calculate the corresponding energies \(E_n\).
- Describe and plot wavefunctions for a finite square well potential. Discuss the significance of a non-zero wavefunction in regions where \(E < U\). Use these ideas to explain quantum mechanical tunneling.
- Explain the band structure that typically describes quantum energy levels in extended systems.
- Relate the emitted or absorbed photon energies, wavelengths and frequencies to the energy levels of various systems. Explain fluorescent and phosphorescent behavior.
Homework
- Friday's Assigned Problems: Supp CH 4: 1, 2, 4, 6, 9, 10, 12, 13, 14, 16
- Monday's Hand-In Problems from Lecture 16:
Supp CH 4: 3, 11, 15, 17, 18
Note: this is only the second half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads.
- Simulation showing the wavefunctions and energy levels for the infinite square well.
Videos of swimming bacteria with GFP mutation near glass surface
The two videos below were taken at Bucknell of bacillus subtilis bacteria swimming near the glass surface with 100X magnification in a Nikon Eclipse inverted microscope. The bacteria have been genetically mutated to contain GFP -- green fluorescent protein -- which cause the bacteria to fluoresce green when illuminated by violet light. The movie on the left shows the bacteria illuminated with epifluorescence, showing not only the bacteria right at the surface (in focus) but also bacteria swimming away from the surface (out of focus). The movie on the right shows the bacteria imaged with TIRF (total internal reflection fluorescence) techniques in which only the parts of the bacteria within approximately 100 nm of the surface are illuminated.
Videos of example problems
To see the problem statement, click on the link below. To play the video example, click on the underlined words "Video Demonstration" near the top of the page with the problem statement.- Long Lensoo example of particle in box, asking both for energy of a state and for probabilities of finding the electron in different regions. (This is a 10-minute video.)
- Lensoo Example showing how to work with absorption and emission processes, using a particle-in-box system. NOTE: Part (a) has a mathematical error -- the 8 in the denominator at the end of the problem shouldn't be there.
- Video example of light emitted from a transition. (This one uses the result for the energy levels of Hydrogen, which we haven't covered yet, but the equation for En is given, so this problem can be done with what you know from today's lecture.)
Pre-Class Entertainment
- Dire Wolf - The Grateful Dead
- Hold On, I'm Comin' - Sam & Dave
- What Would You Say - Dave Matthews Band
- All I Wanna Do - Sheryl Crow
- Late In the Evening - Paul Simon
Assigned Problems Guide
- Supp 4-1: medium-long. Connecting our techniques for waves with node-node end condition to solutions of Schrödinger's equation for the infinite square well. Need to be able to take derivatives and show that $\psi_3(x)$ is a solution.
- Supp 4-2: medium. One trick: use $E_n = h^2 n^2/(8mL)$ and multiply by $c^2/c^2$. Then you can use $hc=1240\,\text{eV$\cdot$nm}$ in the numerator at $mc^2=0.511\,\text{MeV}$ in the denominator, to get your answer in eV.
- Supp 4-4: quick. Sketch the probability density and interpret it.
- Supp 4-6: medium length, but challenging. Examples 3 and show that the wavefunction can extend into regions where $E
- Supp 4-10: medium-long. Go to Example 3 to find $\kappa$. In part (b), the distance $d$ is defined by the very last equation in the statement. (c) and (d) are about why is tunneling an issue in the microscopic world but not so much on our macroscopic scale.
- Supp 4-12: medium-long. First, find the four energy levels $E_1$, $E_2$, $E_3$, and $E_4$. Hint: Find $E_1$ in eV, and then use $E_n = n^2 E_n$ to get the other energies. Then map out all the transitions, i.e., $4\to 3$, $4\to 2$, ... but also $3\to 2$ and $2\to 1$, etc. and find all the $\Delta E$'s, which tells you $E_\text{ph}$. Then convert to photon wavelengths.
- Supp 4-13: medium-long. First, find the photon energy. The use Eq. (4.25) to solve for what radius $R$ is required to get that photon energy.
- Supp 4-14: medium-short. There is some physical context, but in the end, it's just identifying that transition energies equal photon energies, and then finding wavelengths.
- Supp 4-16: medium-short. Same comment as for problem 4-14.