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Hi, I've found the expectation value of Sz, which is hbar/2 (|[tex]\psi[/tex]up|2 - |[tex]\psi[/tex]down|2) by using the formula:
<Si> = <[tex]\psi[/tex]|Si[tex]\psi[/tex]> where i can bex, y or z and [tex]\psi[/tex] is the 'spinor' vector.
I tried to find Sx using the same formula, however, I could only get as far as:
hbar/2 (([tex]\psi[/tex]up)*[tex]\psi[/tex]down + ([tex]\psi[/tex]down)*[tex]\psi[/tex]up)
In my lecture notes, it has the (final) answer (only) as hbar Re{([tex]\psi[/tex]up)*[tex]\psi[/tex]down}.
Similarly, for Sy it gives the expectation value of hbar Im{([tex]\psi[/tex]up)*[tex]\psi[/tex]down}.
I'm not sure how to get from my answer for <Sx> to the one in the notes. I'm assuming it's just a lack of knowledge of some identity with complex numbers.
Any help is appreciated, thanks in advance.
<Si> = <[tex]\psi[/tex]|Si[tex]\psi[/tex]> where i can bex, y or z and [tex]\psi[/tex] is the 'spinor' vector.
I tried to find Sx using the same formula, however, I could only get as far as:
hbar/2 (([tex]\psi[/tex]up)*[tex]\psi[/tex]down + ([tex]\psi[/tex]down)*[tex]\psi[/tex]up)
In my lecture notes, it has the (final) answer (only) as hbar Re{([tex]\psi[/tex]up)*[tex]\psi[/tex]down}.
Similarly, for Sy it gives the expectation value of hbar Im{([tex]\psi[/tex]up)*[tex]\psi[/tex]down}.
I'm not sure how to get from my answer for <Sx> to the one in the notes. I'm assuming it's just a lack of knowledge of some identity with complex numbers.
Any help is appreciated, thanks in advance.