How to Represent |+x> and |-x> Using |+y> and |-y> as Basis?

[danJm]

Homework Statement

Determine the column vectors representing the states |+x> and |-x> using the states |+y> and |-y> as a basis.

Homework Equations

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The Attempt at a Solution

The hint my prof gave us was that since |+x> = 1/√2|+z> + 1/√2|-z> we can eliminate the states |+z> and |-z> in favor of |+y> and |-y>

I'm just lost I guess, I'm not sure how to eliminate the z states in favor of y. Any further hint or suggestion would be much appreciated.

 

[vela]

Think about where |+x> = 1/√2|+z> + 1/√2|-z> came from, and what exactly you mean by |+x>, |-x>, |+y>, |-y>, |+z>, and |-z>.

I assume you're talking about spin 1/2. You never really said.

 

[danJm]

yes, spin 1/2 particles.

The book we're using is Townsend, A Modern Approach to QM. The book starts with spin 1/2 particles and examination of the Stern-Gerlach experiment.

When it's talking about finding the constants (e.g. 1/√2) it says one solution is to choose c₊ and c_- to be real, namely c₊=1/√2 and c_-=1/√2, the more general solution for c₊ and c_- may be written
c+=e^{i[itex]\delta₊[/itex]}/√2 and
c-=e^{i[itex]\delta₊[/itex]}/√2

where [itex]\delta₊[/itex] and [itex]\delta_-[/itex] are real phases that allow for the possibility that c₊ and c_- are complex.

That said, can i just name |+x> in the y basis similarly to it was in the z basis?
It seems kinda vague to me.

 

[vela]

I'm not sure what you mean by "just name |+x> in the y basis similarly to it was in the z basis". There are precise definitions to the basis vectors. Do you know what they are?

 

[danJm]

no, apparently i do not.

 

[vela]

What's their relation to the spin operators S_x, S_y, and S_z?