Trying to work with Susskind's Theoretical Minimum book

Nov 2013
New Zealand

Trying to figure out the above question.

Susskind states the relationship between certain state vectors when using a measuring apparatus to detect spins in certain directions.
He comes up with:

\(\displaystyle |r> = \frac{1}{\sqrt{2}} |u> + \frac{1}{\sqrt{2}} |d> \)
\(\displaystyle |l> = \frac{1}{\sqrt{2}} |u> - \frac{1}{\sqrt{2}} |d> \)

where u,o, l, r, i, o stand for the state vectors up |u>, down |d>, left |l>, right |r>, in |i> and out |o>.

He also states the following experimental results

<r|l> = 0
<l|r> = 0
<u|d> = 0
<d|u> = 0
<i|o> = 0
<o|i> = 0

and if it's an inner product with itself it equals 1 ( <u|u> = 1, <d|d> = 1, etc)

and other combinations comes out to 1/2
<u|o><o|u> = 1/2
<d|o><o|d> = 1/2
<l|i><i|l> = 1/2
<i|l><l|i> = 1/2

With respect to the first exercise I figured out the first part. It was merely taking the inner product <u|i> and <d|i> and same for <u|o> and <d|o> where u,o, l, r, i, o stand for the state vectors up, down, left, right, in and out. It gives me the info that <u|d> = <l|r> = <i|o> = 0 and most of the other combinations = 1/2 when multiplied by their complex conjugate. But I am at a loss to know where to begin to derive \(\displaystyle \alpha \beta^* + \alpha^* \beta\). I can't fathom how the alpha and beta symbols got transposed if we're limiting ourselves to inner products. Any advice?
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