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Lecture 20: Quantum Entanglement
April 9, 2026
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
Updated Problem 9: Suppose we modified Bell's experiment and aligned the positron detector to be at a $60^\circ$ angle with respect to the $z$-axis.
- In quantum mechanics, if the electron is measured to be spin down ($S^\text{elec}_z=-\hbar/2$), what is the probability that the positron will be found to have $S^\text{pos}_{60^\circ} =+\hbar/2$?
- Now we turn to hidden variable theories. Bell's theorem states that for all possible hidden variable theories, if the electron is measured to be spin down ($S_z^\text{elec}=-\hbar/2$), then the probability that the positron will be measured spin up must obey the inequality $\text{Prob}(S_\theta^\text{pos}=+\hbar/2)\leq 1 - \frac{\theta}{180^\circ}$. For this $60^\circ$ rotated detector, calculate the upper bound on this positron probability.
- In this case, are the predictions of quantum mechanics and hidden variable theories compatible? Why or why not?
- Monday's Hand-In Problems from Lecture 20:
Supp CH 8: 11, 12, 13, 14, 15
Updated Problem 15: Suppose we modified Bell's experiment and aligned the positron detector to be in the $+x$ direction. That is, we will measure the $S_x$ value for the positron.
- In quantum mechanics, if the electron is measured to be spin down ($S^\text{elec}_z=-\hbar/2$), what are the probabilities that the positron will be found to have $S_x=\pm\hbar/2$?
- Now we turn to hidden variable theories. Bell's theorem states that for all possible hidden variable theories, if the electron is measured to be spin down ($S_z^\text{elec}=-\hbar/2$), then the probability that the positron will be measured spin up must obey the inequality $\text{Prob}(S_\theta^\text{pos}=+\hbar/2)\leq 1 - \frac{\theta}{180^\circ}$. For this positron detector oriented in the $+x$ direction, calculate the upper bound on this positron probability $\text{Prob}(S_x^\text{pos}=+\hbar/2)$.
- In this case, are the predictions of quantum mechanics and hidden variable theories compatible? Why or why not?
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
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
- Soul Man - Sam & Dave
- Bad Reputation - Joan Jett