How does measuring spin in arbitrary direction relate to eigenstate equations?

[Muthumanimaran]

I am currently reading Prof.Leonard Susskind's Lecture on Quantum Mechanics. In the Chapter: Spin in the arbitrary directions, in the subdivision Eigenstates
In case $$\lambda=1$$
Prof states that measuring spin in arbitrary +n state gives me +1 as eigenvalue, what I don't understand is the next expression $$n_zα+n_−β=α$$
I have no idea how this expression comes here, please help me. The link to the lecture is given below:
http://www.lecture-notes.co.uk/suss...ments/lecture-4/spin-in-arbitrary-directions/

 

[Paul Colby]

There are two equations for $\alpha$ and $\beta$ given by the eigenvalue equation. This is the one for $\alpha$. The other is for $\beta$.

 

[vanhees71]

I don't see any obvious mistakes (I've not followed everything thoroughly). All that Suskind in fact does is to diagonalize the matrix $\vec{n} \cdot \vec{\sigma}$. Where's your specific problem?

 

[Muthumanimaran]

There are two equations for $\alpha$ and $\beta$ given by the eigenvalue equation. This is the one for $\alpha$. The other is for $\beta$.

I don't how that's lead to the expression $$n_zα+n_−β=α$$

 

[Paul Colby]

$\sigma_n$ times the vector $(\alpha,\beta)$ gives the same vector multiplied by 1. The equation follows from the top row of the matrix.

 

[Muthumanimaran]

ok, thank you, now I understand how it comes. Initially I misunderstood the Lecture. Its pretty straightforward.