Trying to work with Susskind's Theoretical Minimum book

Nov 2013
552
30
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:

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

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
et

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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