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Spin, let me make sure I have this straight

  1. Jun 4, 2010 #1
    So if you measures the spin of an electron in a Hydrogen atom (which I understand requires a magnetic field to eliminate the degeneracy of energy levels) in an arbitrary direction

    [tex] x^i[/tex]

    you would measure plus or minus

    [tex]\frac{eh}{2Mc}B_i[/tex]

    in Joules. Where [tex]B_i[/tex] is the magnetic field component in the direction [tex] x^i[/tex]

    I understand you never measure anything between this.

    What about the magnetic field components of the other 2 orthogonal dimensions? Is the spin in those directions also

    [tex]\frac{eh}{2Mc}B_i[/tex]

    That would give a total intrinsic angular momentum of 3/2.
     
  2. jcsd
  3. Jun 5, 2010 #2
    Well it's been a couple years since I took my quals, so someone correct me if I'm wrong. But I believe the Hamiltonian for this system is:

    [tex]H=\dfrac{e\hbar}{2mc}B_z\sigma_z[/tex]

    But if you want to find the possibile energies that you'd get by making a measurement of the x-component of the spin, you're basically finding the eigenvalues of the x spinor. So you'd do,

    [tex]\dfrac{e\hbar}{2mc}B_z\sigma_z \dfrac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\end{array}\right) = E \dfrac{1}{\sqrt{2}}\left(\begin{array}{cc}1\\1\end{array}\right)[/tex]

    So assuming I got that right, now you just need to do the eigenvalue problem.
     
  4. Jun 5, 2010 #3

    alxm

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    You can't sum up the components; the operators don't commute for different directions.
    They do commute with S2 and so you get an expectation value [tex]S^2 = 3\hbar^2/4[/tex] and more generally [tex]S = \sqrt{s(s+1)}[/tex]

    So the total spin angular momentum is [tex]S = \hbar\frac{\sqrt{3}}{2}[/tex]
     
  5. Jun 6, 2010 #4
    So what does it mean to say the spin of an electron is l=1/2 where l is the total angular momentum.

    Edit:

    I think I understand my mistake, l is the total angular momentum quantum number, not the energy eigenvalue.

    So then am I correct in saying that the absolute value of the intrinsic angular momentum of a particle is equal in any direction a magnetic field of equal intensity is applied?
     
    Last edited: Jun 6, 2010
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