Calculate a spin state as a function of the base spin states

[Luke_Mtt]

TL;DR Summary

Hi everyone,
I recently started approaching for the first time to the quantum mechanics field by reading the book of Susskind and Friedman "Quantum Mechanics, The Theoretical Minimum" but I couldn't figure how the authors can express some spin state in function of two (orthogonal) base spin states (II chapter).

Practically it is said that, given two spin states |u⟩ (up) and |d⟩ (down) - which are the spin measured along the +z and -z semiaxes - such that they are orthogonal ( ⟨u|d⟩ = ⟨d|u⟩ = 0), it is possible to write any other spin states using a linear combination of these two (because they are a base of the states' vector space). So it is:

|A⟩ = α_u |u⟩ + α_d |d⟩ = ⟨u|A⟩ |u⟩ + ⟨d|A⟩ |d⟩

It is then said that given a spin state |A⟩, the probability to measure up or down (|u⟩ and |d⟩) spin states are:

P_u = ⟨A|u⟩ ⟨u|A⟩ = ⟨u|A⟩* ⟨u|A⟩ = α_u* α_u
P_d = ⟨A|d⟩ ⟨d|A⟩ = ⟨d|A⟩* ⟨d|A⟩ = α_d* α_d

(where x* is the complex conjugate)
Then, the total probability is 1 so that:

α_u* α_u + α_d* α_d = 1.

Which is equivalent to say that |A⟩ is normalized to the unit (⟨A|A⟩ = 1)

Then it is introduced another vector cuple considered as another base. This two vectors, |r⟩ (right) and |l⟩ (left) represent the spin state along the +x and -x semiaxes. Quoting verbatim the book it is said that "If a spin state |A⟩ prepares the initial state |r⟩, and an instrument that stores the spin value is rotated (from |r⟩ because it has prepared it) to measure the z axe's spin (σ_z), the probability to obtain up or down is the same (because the x axe is perpendicular to the z one), so it must be: α_u* α_u = α_d* α_d = ½". Then the authors state that a vector which satisfy this conditions is, for example:

|r⟩ = α_u |u⟩ + α_d |d⟩ = ⟨u|r⟩ |u⟩ + ⟨d|r⟩ |d⟩ = 1/√2 |u⟩ + 1/√2 |d⟩

After, they say, in addition to a phase ambiguity (please, can you make me understand this point too, because I couldn't figure out what phase ambiguity is), |r⟩ and |l⟩ must satisfy ⟨l|r⟩ = ⟨r|l⟩ = 0 too because of the fact of being a base of the vector space. They then say that this conditions bind the |l⟩ vector to take the form:

|l⟩ = α_u |u⟩ + α_d |d⟩ = ⟨u|l⟩ |u⟩ + ⟨d|l⟩ |d⟩ = 1/√2 |u⟩ - 1/√2 |d⟩

Then they repeat almost the same procedure for the y axe. So, calling |i⟩ (in) and |o⟩ (out) the spin state along the +y and -y semiaxes it is possible to use this two spin state as a base, provided that ⟨i|o⟩ = ⟨o|i⟩ = 0. Then it is known that:

α_u* α_u = ⟨u|o⟩* ⟨u|o⟩ = ⟨o|u⟩ ⟨u|o⟩ = ½ // α_u* α_u in function of o
α_d* α_d = ⟨d|o⟩* ⟨d|o⟩ = ⟨o|d⟩ ⟨d|o⟩ = ½ // α_d* α_d in function of o
α_u* α_u = ⟨u|i⟩* ⟨u|i⟩ = ⟨i|u⟩ ⟨u|i⟩ = ½ // α_u* α_u in function of i
α_d* α_d = ⟨d|i⟩* ⟨d|i⟩ = ⟨i|d⟩ ⟨d|i⟩ = ½ // α_d* α_d in function of i

And also because y is orthogonal both at z and at x it mus be

α_r* α_r = ⟨r|o⟩* ⟨r|o⟩ = ⟨o|r⟩ ⟨r|o⟩ = ½ // α_r* α_r in function of o
α_l* α_l = ⟨l|o⟩* ⟨l|o⟩ = ⟨o|l⟩ ⟨l|o⟩ = ½ // α_l* α_l in function of o
α_r* α_r = ⟨r|i⟩* ⟨r|i⟩ = ⟨i|r⟩ ⟨r|i⟩ = ½ // α_r* α_r in function of i
α_l* α_l = ⟨l|i⟩* ⟨l|i⟩ = ⟨i|l⟩ ⟨l|i⟩ = ½ // α_o* α_o in function of i

So, they say, apart from the phase ambiguity, the vectors for |i⟩ and |o⟩ which satisfies this conditions are:

|i⟩ = α_u |u⟩ + α_d |d⟩ = ⟨u|i⟩ |u⟩ + ⟨d|i⟩ |d⟩ = 1/√2 |u⟩ + i/√2 |d⟩
|l⟩ = α_u |u⟩ + α_d |d⟩ = ⟨u|o⟩ |u⟩ + ⟨d|o⟩ |d⟩ = 1/√2 |u⟩ - i/√2 |d⟩

Finally, the thing I couldn't understand, apart from the phase ambiguity, is "how the authors came up with the α_u and α_d coefficients for |r⟩, |l⟩, |i⟩, |o⟩"? Did they already know that this coefficients would have worked with that specific conditions or there is a process behind this choice? If it is, which calculus do I have to do to derive these coefficients?

P.S. My english is not perfect, so sorry for eventual errors.

Thank you for your time!

 

[atyy]

The phase ambiguity means that you can multiply both coefficients by $\exp(i\theta)$, where $\theta$ is any angle.

I haven't understood your second question about the coefficients. Do you mean you don't understand how they derived that $a_{u} = \frac{1}{\sqrt{2}}$?

 

[Luke_Mtt]

Exactly. Because the authors gave the final equations by saying "this vector can satisfy this particular conditions". But both in |r⟩, |l⟩ and |i⟩, |o⟩ I didn't understand how they have derived the α_u and α_d coefficients for this four spin states.