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Lecture 20: Quantum Entanglement
April 10, 2025
Reading Assignment
- Read: Supplementary Reading Ch 8
Objectives
- (Continuing objective) Describe applications of the concepts of quantum mechanics to everyday “real-life” situations.
- Identify whether a given two-particle state is separable or entangled. For separable states, obtain the separate single-particle states.
- For a given two-particle state, calculate the probabilities associated with a particle-2 measurement given the result of a particle-1 measurement.
- Describe the EPR paradox and Bell's inequality and their implications for locality and the completeness of quantum states.
Homework
- Friday's Assigned Problems: Supp CH 8: 1, 2, 3, 4, 5, 8, 9, 10
- Monday's Hand-In Problems from Lecture 20:
Supp CH 8: 11, 12, 13, 14, 15
Note: this is only the second half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads. Answers: CT1 - 5; CT2 - 6; CT3 - 3; CT4 - 6; CT5 - 5
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.- Entanglement Example #1. This is very similar to Supp 8-3 -- rewriting a two-particle state for an electron and a positron. (NOTE: there is an error where he accidentally writes "2/3" instead of "2/35" for about 5 minutes, but later catches the error and fixes it.)
- Entanglement Example #2 (continuation of #1). This uses the same state as example number 1, but addresses the question of whether or not the state is entangled, specifically showing that a measurement of the electron changes the probabilities for outcomes of a measurement of the positron spin.
Pre-Class Entertainment
- You Learn - Alanis Morissette
- Pump It Up - Elvis Costello
- Blitzkrieg Bop - The Ramones
- Ohio - Crosby, Stills, Nash, and Young
- Bad Reputation - Joan Jett
Assigned Problems Guide
- Supp 8-1: medium-quick. Using our rule of $|c_n|^2$ for the probabilities, while interpreting the two-particle states.
- Supp 8-2: medium-long. Algebra. Use normalization to find the $c_+$ and $c_-$ coefficients, and then work out $|\phi_1\rangle$ and $|\phi_2\rangle$.
- Supp 8-3: medium-long. Algebra again. Factor out the electron spin part, and then find the $c_+$ and $c_-$ coefficients by normalization.
- Supp 8-4: medium. For (a) you need to go the other way. Multiply out the factored state to find the coefficients of states where the positron is spin up. Parts (b)-(d) are quicker, using state collapse.
- Supp 8-5: medium-quick. Trick: just take a valid electron state and a valid positron state and them multiply them together (like FOIL).
- Supp 8-8: medium-quick, interpretation question.
- Supp 8-9: medium-long. Use Eq. 8.18 with the $b_+$ and $b_-$ coefficients in part (a). For part (b), it's more a geometrical interpretation like in Figure 8.3.
- Supp 8-10: medium-quick interpretation question.