Spin

Electron Spin

Spin

• in mechanics, we defined angular momentum as:

  • $\vec{L}=\vec{r}\times \vec{p}$

  • A charged particle orbiting a point has angular momentum, and exhibits a magnetic dipole moment.

    • we can now say since it has this “spin” it would have also a moment because of it
      • ${\mu }_{}$

• This effects dipole electron gun guy

  • Classical

    • ${\mu }_z=|\mu |\cos \theta$
      • it’s like a gradient

  • Result

    • ${\mu }_s=-g_s\frac{e}{2m_e}S$
      • because of the electron spin it “quantizes” and only goes up or down no gradient

      • What is S

        • it’s the spin
        • $\text{ when the state is up }|up>=+\frac{\hslash }{2}$
        • $\text{ when stat is down }|down>=-\frac{\hslash }{2}$

• Spin has two possible states

  • The states are in super position

    • Dirac Notation
    • $|spin>=a|up>+b|down>$ with |a|^2+|b|^2 =1
      • with the probability of that we now that
        • $P_{up}=|<up|spin>{|}^2=|a{|}^2$

• Stern Gerlach Device

  • When measuring spin you can’t know the spin in a direction while knowing the other

  • We can use Born Rule to determine possible results and probabilities

    • Example

      • $|\psi >=\frac{1}{2}|up>+\frac{i\sqrt{3}}{2}|down>$
        • we must make sure the vector is normalized by multiplying by the complex conjugate

Spin Matrices

• We derived spine operators $S_x,S_y,\text{and }{S}_z$ by pauli matrices

  • $S_i\equiv \frac{\hslash }{2}{\sigma }_i,{\sigma }_x=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
    • then we write each of these as a basis of eachother to be able to use Born Rule

  • When S is not a axis we use

    • $S_{\theta }=\frac{\hslash }{2}\left[\begin{array}{cc}\cos \theta & \sin \theta \\ sin\theta & -\cos \theta \end{array}\right]$

• Example

  • Spin observable

    • show that $p_{up}=|<up|ccw>{|}^2=\frac{1}{2}$

Global vs relative phases

• For generic 2 level quantum states we have $|\psi >=a|up>+b|down>$

• we can discover two types of phase:

  • Global phase: applies to entire quantum state

    • phase can be factored out

    • they are unobservable and has no influence on measurement outcomes

  • relative phase: difference in phase between components of the state

    • phase is difference between the two combinations
      • we can remove the relative phase then we can find the relative phase differences (smallest difference like velocity 75m/s vs 50m/s we can take out 50 and the difference is 25)