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6.P: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

  1. Calculate the Clebsch-Gordon coefficients for adding spin one-half to spin one.
  2. Calculate the Clebsch-Gordon coefficients for adding spin one to spin one.
  3. An electron in a hydrogen atom occupies the combined spin and position state whose wavefunction is
    ψ=R21(r)[1/3Y10(θ,φ)χ++2/3Y11(θ,φ)χ].
    1. What values would a measurement of L2 yield, and with what probabilities?
    2. Same for S2 .
    3. Same for J2 .
    4. Same for r , φ ?
    5. What is the probability density for finding the electron in the spin up state (with respect to the r ?
  4. In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by
    $ V({\bf x}) = V_1(r) + V_2(r)\left[3\,\frac{(\mbox{\boldmath $\sig...
...\right] + V_3(r)\,\mbox{\boldmath $\sigma$}_n\cdot\mbox{\boldmath $\sigma$}_p,
$
    where σ σ V(x) with respect to the overall spin state.]
  5. Consider two electrons in a spin singlet (i.e., spin zero) state.
    1. If a measurement of the spin of one of the electrons shows that it is in the state with z -component of the spin of the other electron yields Sy=/2 , what is the probability that a measurement of the Sx=/2 ?
    2. Finally, if electron 1 is in a spin state described by $ \cos\alpha_1\,\chi_+
+ \sin\alpha_1\,{\rm e}^{\,{\rm i}\,\beta_1}\,\chi_-$ , and electron 2 is in a spin state described by $ \cos\alpha_2\,\chi_+
+ \sin\alpha_2\,{\rm e}^{\,{\rm i}\,\beta_2}\,\chi_-$ , what is the probability that the two-electron spin state is a triplet (i.e., spin one) state?

Contributors

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 6.P: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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