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I have a question that i think belongs here or in the linear algebra section; but now i put it here, maybe someone will move the post if it's in the wrong place.

The question:

I have a normalized (1D) state in the x-axis in the harmonic oscillator

[tex]\Psi(x,t=0)=1/\sqrt{45}(6u_0(x)\chi_+ +(2+i)u_1\chi_- -2u_1(x)\chi_+[/tex]

where [tex]u_n[/tex] is the n'th normalized eigenstate of the 1D HO and [tex]\chi_{+/-}[/tex] are the normalized eigenspinors in z-direction. Now i must find the possibility of getting the result +h/2 when i measure [tex]S_y[/tex]. My idea was to take the inner product with the desired state and square the coefficient:

[tex]prob(h/2)=(\left\langle \chi^y_+ | \Psi\right\rangle)^2[/tex]

but how do i do that when i have both spinors(vectors) and functions of x at the same time? If anyone could do the calculation stepwise, I would appreciate it very much!

Best regards

Oistein