Representation of spin matrices

  • Thread starter bznm
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  • #1
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I have just started to study quantum mechanics, so I have some doubts.

1) if I consider the base given by the eigenstates of s_z [tex]s_z | \pm >=\pm \frac{\hbar}{2} |\pm>[/tex] the spin operators are represented by the matrices

[tex]s_x= \frac {\hbar}{2} (|+><-|+|-><+|)[/tex]
[tex]s_y= i \frac {\hbar}{2}(|-><+|-|+><-|)[/tex]
[tex]s_z=\frac{\hbar}{2}(|+><+|-|-><-|)[/tex]

but I don't have clear idea of what they correspond to concretely. Considering the Dirac formalism, can they be represented by Pauli's matrices?

[tex]s_x=\frac {\hbar}{2} \sigma_x; s_y= i \frac {\hbar}{2} \sigma_y; s_z=\frac {\hbar}{2} \sigma_z[/tex]

2) I can write a vector using its components, e.g. v=(a,b)
but which are the components of the eigenstates of s_z?

3) If I have a state such as [tex]s_x |\phi>=\frac {\hbar}{2} |\phi> [/tex]
and I want to write it using the base of the eigenstates of s_z,
can I write [tex] |\phi>=a|+>+b|->[/tex], with [tex]|a|^2+|b|^2=1?[/tex] (I need this condition to have a normalized vector)
Is it equal to [tex]\frac {e^{i\theta}}{\sqrt 2}(|+>+|->)?[/tex]

Many thanks for your help!
 

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  • #2
Orodruin
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Considering the Dirac formalism, can they be represented by Pauli's matrices?
Yes.

2) I can write a vector using its components, e.g. v=(a,b)
but which are the components of the eigenstates of s_z?
Yes, if you use the representation of the spin operators in terms of Pauli matrices.

can I write
Yes, what you quoted here was a general spin state. You need to fix a and b so that it is actually an eigenvector of s_x.



Is it equal to
... Which you did here. The exponential is an arbitrary phase factor and can be dropped.
 
  • #3
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Thanks a lot!
 

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