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6.P: Exercises

  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

     

    \[\psi = R_{2\,1}(r)\,\left[\sqrt{1/3}\,Y_{1\,0}(\theta,\varphi)\,\chi_+ + \sqrt{2/3}\,Y_{1\,1}(\theta,\varphi)\,\chi_-\right].\]

     

    1. What values would a measurement of $ L^2$ yield, and with what probabilities?
    2. Same for $ L_z$ .
    3. Same for $ S^2$ .
    4. Same for $ S_z$ .
    5. Same for $ J^{\,2}$ .
    6. Same for $ J_z$ .
    7. What is the probability density for finding the electron at $ r$ , $ \theta$ , $ \varphi$ ?
    8. What is the probability density for finding the electron in the spin up state (with respect to the $ z$ -axis) at radius $ r$ ?

     

  4. In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by

     

    $\displaystyle 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 $ r=\vert{\bf x}\vert$ , $ \sigma$ $ _n$ denotes the vector of the Pauli matrices of the neutron, and $ \sigma$ $ _p$ denotes the vector of the Pauli matrices of the proton. Calculate the potential energy for the neutron-proton system:
    1. In the spin singlet (i.e., spin zero) state.
    2. In the spin triplet (i.e., spin one) state.
    [Hint: Calculate the expectation value of $ V({\bf 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 $ S_z=\hbar/2$ , what is the probability that a measurement of the $ z$ -component of the spin of the other electron yields $ S_z=\hbar/2$ ?
    2. If a measurement of the spin of one of the electrons shows that it is in the state with $ S_y=\hbar/2$ , what is the probability that a measurement of the $ x$ -component of the spin of the other electron yields $ S_x=-\hbar/2$ ?
    3. 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?

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