I have done problem 4.50 from Griffiths' Quantum Mechanics (2nd ed), quoted below, and got the correct answer. What would be a physical situation where this would be useful?
I have already had some people tell me to look at Bell's Theorem in Ch 12, which I did. But that is seven chapters ahead of where I am at in the book, and contains things that are over my head at this point. Can anyone tell me, in plain undergrad terms, a physical situation where this would be useful?
Problem 4.55 Suppose two spin-$1/2$ particles are known to be in the singlet configuration (Equation 4.178). Let $S_a^{(1)}$ be the component of the spin angular momentum of particle number 1 in the direction defined by the unit vector $\hat{a}$. Similarly, let $S_b^{(2)}$ be the component of 2's angular momentum in the direction $\hat{b}$. Show that $$\langle S_a^{(1)} S_b^{(2)} \rangle = -\frac{\hbar^2}{4}\cos\theta,\tag{4.198}$$ where $\theta$ is the angle between $\hat{a}$ and $\hat{b}$.
Answer
I see you're studying Griffiths' book on Quantum Mechanics. It is interesting that you're asking about this particular exercise, since this is an equation that Griffiths is cross-referencing in the Afterword (Bell's Theorem), which I clearly suggest you go and read, specifically about the Bell inequalities.
This exact problem of orientation selection is what is necessary to do a specific calculation for the proposed experiment to try and probe the (then new) elements of quantum entaglement.
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