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Okay so I'm pretty dumb and slightly struggling in quantum mechanics this semester.
Is you have some measurement that measures a 1/2-spin system, the component along the z-axis comes through as:
|Ψ> = |+> + |->
But then why is the component along the x and y axis any different? For example my book defines x and y as (by convention):
x: |+> = 1/sqrt(2) [ |+> + |-> ]
x: |-> = 1/sqrt(2) [ |+> - |-> ]
and
y: |+> = 1/sqrt(2) [ |+> + i|-> ]
y: |-> = 1/sqrt(2) [ |+> - i|-> ]
What do these axes results mean? Is it implying that you first took a measurement with respect to the z-axis, THEN measured either x or y? I'm having some trouble conceptualizing what these x and y equations are telling you.
That's all. Thanks!
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>>9168513
>What do these axes results mean?
It's just a change in basis.
>Is it implying that you first took a measurement with respect to the z-axis, THEN measured either x or y?
No, if you measure with respect to the z-axis you would get |+z> or |-z>. Lets say |+z>. Then the probably the 2nd measurement wrt the x axis will be positive will be <+x|+z>^2=0.5 and will be negative is <-x|+z>^2 =0.5.
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>>9168535
>>9168535
>No, if you measure with respect to the z-axis you would get |+z> or |-z>. Lets say |+z>. Then the probably the 2nd measurement wrt the x axis will be positive will be <+x|+z>^2=0.5 and will be negative is <-x|+z>^2 =0.5.
Right, makes sense and lines up with the thought experiments (stern gerlach).
But for some reason I still can't wrap my head around why these are different. I understand they're a change in basis, orthogonal, and all that jazz, but for some reason I can't conceptualize it...
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>>9168574
To convince yourself of this, calculate the eigenvectors of the pauli spin matrices for a spin -1/2 particle. For example:
Sx |+->=j|+-> implies that mat (-j,h/2,h/2,-j)=0.
Det (mat (-j,h/2,h/2,-j))=0 implies that j=+-h/2.
Forj=+h/2, mat (-1,1,1,-1)×mat (x1,x2)=0 implies that x1=x2. Since x1^2+x2^2=1, 2x1^2=1 implies x1=+-1/(2)^2. This means that the spinor is (x1,x2)=(1/(2)^2,1/(2)^2).
For j=-h/2, the eigenst in or comes out to be (1/(2)^2,-1/(2)^2).
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>>9168668
Well the pauli matricies already imply something I don't understand. Not sure how I can start with something I don't understand and work back to confirm the thing I don't understand.
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