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Lecture 19: Three-Dimensional Wavefunction and Semiconductors
April 8, 2025
Reading Assignment
- Read: Supplementary Reading Ch 7
- Skim: Section 7.5.1
Objectives
- (Continuing objective) Describe applications of the concepts of quantum mechanics to everyday “real-life” situations.
- Test solutions for the hydrogen atom in the three-dimensional Schrödinger Equation.
- Write down and relate the possible values of the principal quantum number \(n\), the orbital quantum number \(l\), and the magnetic quantum number \(m_l\) for states of an electron in the hydrogen atom. Calculate the energy \(E_n\), angular momentum magnitude \(|\vec{L}|\), and \(z\)-component of angular momentum \(L_z\) using these quantum numbers.
- Relate the conductivity (or lack thereof) of a solid to the band structure of its energy levels. Use simple “\(k_BT\)” approximations to determine how the conduction properties depends on the temperature.
- Explain doping and describe “n-type” and “p-type” semiconductors. Explain how p-n junctions produce diode and transistor behavior, concentrating in particular on the behavior of the depletion zone. Explain how light-emitting diodes work, and calculate the wavelength/frequency of LED emission, based on the band structure of the semiconductor.
Homework
- Wednesday's Assigned Problems: Problem X13; Supp CH 7: 2, 3, 4, 9, 11, 12, 13, 15
- Monday's Hand-In Problems from Lecture 19:
Supp CH 7: 5,
8, 10, 14, 16Note: this is only the first half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads.
- Hydrogen Atom Viewer
- Band structure simulation
- Movie showing a forward-biased diode. NOTE: the ``DZ'' label should not be there, since the depletion zone no longer exists when forward biased.
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.- Example of working with n, l, ml and ms stuff. The first half of this example is really an explanation for the first two rows of the Periodic Table of the Elements. The second half is a question about how many electrons can fill the n = 5 energy state.
- Video example of light emitted from a transition.
Pre-Class Entertainment
- Tumbling Dice - The Rolling Stones
- Juke - Little Walter
- Feelin' Alright - Joe Cocker
- Pressure Drop - Toots and the Maytals
- Big Chief - Professor Longhair
Assigned Problems Guide
- X13: medium. Online problem (link above). Have the app running in one window and the problem statement in a second window and carefully go through the steps. This will help you understand why a p-n junction acts as a diode.
- Supp 7-2: medium-long. Listing all the $(n,\ell,m_\ell, m_s)$ states can take a little while, but it's worth working through it to drill the concept.
- Supp 7-3: quick. Look up how angular momentum $|\vec L|$ and the quantum number $\ell$ are related.
- Supp 7-4: medium-quick. Again uses relation between $|\vec L|$ and $\ell$, and also uses the relationship between $n$ and $\ell$ (that $\ell$ has to be less than $n$).
- Supp 7-9: medium-quick. Like the example in class with silicon: compare $k_BT$ to the gap energy.
- Supp 7-11: medium. The photon is emitted by a transition from the bottom of the conduction band to the top of the valence band. What is the name for this gap? It rhymes with "map" energy...
- Supp 7-12: medium. Interesting question about how LED lights are manufactured. Remember with fluorescence that we always get how lower energy photons than we send in.
- Supp 7-13: medium. Make sketches of the doped semiconductors and the DZ. Then think about which way to apply $\vec E$ to get the DZ to disappear, or to make it grow larger.
- Supp 7-15: long. First notice that all the $\theta$ and $\phi$ derivatives are gone. So you just need to calculate the $r$ derivative terms, but it's not simply a second derivative. Just follow the steps. At the end, you'll have some powers of $r$ and an exponential of $r$. Cancel what you can, and then you should be left with two constant terms and two $1/r$ terms. Choosing the value for $a_0$ and $E$ can make that work out as a solution.