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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 yield, and with what probabilities?
2. Same for .
3. Same for .
4. Same for .
5. Same for .
6. Same for .
7. What is the probability density for finding the electron at , , ?
8. What is the probability density for finding the electron in the spin up state (with respect to the -axis) at radius ?

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

where , denotes the vector of the Pauli matrices of the neutron, and 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 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 , what is the probability that a measurement of the -component of the spin of the other electron yields ?
2. If a measurement of the spin of one of the electrons shows that it is in the state with , what is the probability that a measurement of the -component of the spin of the other electron yields ?
3. Finally, if electron 1 is in a spin state described by , and electron 2 is in a spin state described by , what is the probability that the two-electron spin state is a triplet (i.e., spin one) state?