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Sparky_

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- TL;DR Summary
- From Dr. Leonard Susskind's Stanford Lecture: Quantum Entanglement, Lecture 4, he sets up a "given particle is spin up along n (arbitrary direction) and discusses : what is probability we measure up along another arbitrary m direction

From Dr. Leonard Susskind's Stanford Lecture: Quantum Entanglement, Lecture 4, he sets up a "given particle is spin up along n (arbitrary direction) and discusses : what is probability we measure up along another arbitrary m direction

He does all of the setup, - calculates the eigenvectors and gives the final answer: ##\frac{1}{2}(1-\cos(\theta))##

Months and months ago I took a stab at the work and gave up - got really messy

Few days ago I thought (for fun) I would tackle it again, jumping to the end I ended up with:

$$\frac{1}{2}((1+\cos(\theta) ) (1 - m_3^2 -n_3^2 + (m_3n_3)^2)$$

$$= \frac{1}{2} (1+\cos(\theta))(m_3^2-1)(n_3^2-1)$$

The work is many pages of algebra with m's and n's, fortunately a lot canceled and then others grouped to simplify

My current result seems clean but has the terms ##(m_3^2-1)(n_3^2-1)## (with##1-\cos(\theta)## factored out

I'm hoping you tell me there is a step to cancel or something

The initial matrices before the turn-the-crank work is:

$$\begin{pmatrix}\sqrt{ \frac{1+m_3} {2} } && \frac{1-m_3}{m+}\sqrt{\frac{1+m_3}{2}} \end{pmatrix}\begin{pmatrix}\sqrt{\frac{1+n_3}{2}} \\ \frac{1-n_3}{n-}\sqrt{\frac{1+n_3}{2}}\end{pmatrix}$$

Thanks

Sparky_

He does all of the setup, - calculates the eigenvectors and gives the final answer: ##\frac{1}{2}(1-\cos(\theta))##

Months and months ago I took a stab at the work and gave up - got really messy

Few days ago I thought (for fun) I would tackle it again, jumping to the end I ended up with:

$$\frac{1}{2}((1+\cos(\theta) ) (1 - m_3^2 -n_3^2 + (m_3n_3)^2)$$

$$= \frac{1}{2} (1+\cos(\theta))(m_3^2-1)(n_3^2-1)$$

The work is many pages of algebra with m's and n's, fortunately a lot canceled and then others grouped to simplify

My current result seems clean but has the terms ##(m_3^2-1)(n_3^2-1)## (with##1-\cos(\theta)## factored out

I'm hoping you tell me there is a step to cancel or something

The initial matrices before the turn-the-crank work is:

$$\begin{pmatrix}\sqrt{ \frac{1+m_3} {2} } && \frac{1-m_3}{m+}\sqrt{\frac{1+m_3}{2}} \end{pmatrix}\begin{pmatrix}\sqrt{\frac{1+n_3}{2}} \\ \frac{1-n_3}{n-}\sqrt{\frac{1+n_3}{2}}\end{pmatrix}$$

Thanks

Sparky_

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