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Lecture 17: Quantum States and Spin
April 1, 2025
Reading Assignment
- Read: Supplementary Reading Ch 5
Objectives
- (Continuing objective) Describe applications of the concepts of quantum mechanics to everyday “real-life” situations.
- Express the state of a quantum system in terms of state vectors. Explain what is meant by the state being “normalized.”
- Explain what is meant by a measurement on a quantum state and the meaning of “collapse of the state.” Given a superposition state, determine the possible results of a measurement of, for example, energy or a spin component and compute the probabilities of obtaining various possible results.
- Describe aspects of quantum mechanical “spin” that can't be explained classically.
- Given a spin state written in one basis, write the same spin state in a different basis. For example, given a spin \(|\text{$\psi$}\rangle\) state in the \(|\text{$\pm z$}\rangle\) basis, be able to express the same state in the \(|\text{$\pm x$}\rangle\) basis.
Homework
- Wednesday's Assigned Problems: A74; Supp CH 5: 1, 3, 5, 9, 10, 11, 14, 17, 20
- Monday's Hand-In Problems from Lecture 17:
Supp CH 5: 4, 12, 13, 15, 19
Note: this is only the first half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads. Answers: CT1 - 6; CT2 - 1; CT3 - 1; CT4 - 4; CT5 - 3
- Whole Brain Atlas -- MRI scans of the brain.
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.- Video example: Dealing with a superposition state. NOTE: when this example was made (a few years ago), we didn't take into account the possibility of complex coefficients (i.e., with ``$i$`'' in them). So, in addition to $\sqrt{3/4}$ for $b$, $b$ could also be $-\sqrt{3/4}$, $i\sqrt{3/4}$, or $-i\sqrt{3/4}$.
- Video Example: Another example with this state stuff. As with the second example above, there are actually a few different possibilities for the constant a.
- Simple example of finding resonant frequency to flip a proton (NMR example).
- Video Example: Magnetic Resonance. This one also looks at spatial variation, something that can be used in MRI devices to probe particular parts of a patient's body.
Pre-Class Entertainment
- The Times They are a-Changin' - Bob Dylan
- I Feel the Earth Move - Carole King
- Banana Pancakes - Jack Johnson
- What's So Funny 'Bout Peace, Love, & Understanding - Elvis Costello
- See Saw - Aretha Franklin
Assigned Problems Guide
- A74: medium. Fun to imagine trying to live in the quantum world!
- Supp 5-1: medium. Turn on your creative juices!
- Supp 5-3: quick. Testing the basic rules of states and probabilities.
- Supp 5-5: long. Need to express the $S_z$ states in terms of the $S_y$ states using Table 5.1, and then collect the $|\text{$+$}y\rangle$ terms.
- Supp 5-9: medium. You can do algebra with the states just like they are variables, so you're solving two equations for the two unknowns of $|\text{$+$}z\rangle$ and $|\text{$-$}z\rangle$
- Supp 5-10: medium-long. Part (a) is straightforward use of $|c_n|^2$ for probability. Part (b) is another substitution problem, like problem 5-5.
- Supp 5-11: medium-long. Yet another substitution problem. Maybe skip this at first and come back to it later for the practice!
- Supp 5-14: medium-quick and very, very cool! Uses our general emitting a photon relation $E_\text{ph}=|\Delta E|$. This low energy transition due to a spin flip in hydrogen creates a very dominant signal throughout the universe!!
- Supp 5-17: medium. Need to bring together all the state and probability tools, and calculate magnitude squares with complex conjugates.
- Supp 5-20: medium. You're being told indirectly the probabilities of measuring $S_z$ to be spin up and spin down. So work out those probabilities, and then see about state coefficients that would give you those probabilities, similar to Concept Test 2.