# 6.P: Exercises

- Calculate the Clebsch-Gordon coefficients for adding spin one-half to spin one.
- Calculate the Clebsch-Gordon coefficients for adding spin one to spin one.
- An electron in a hydrogen atom occupies the combined spin and position state whose wavefunction is \[\psi = R_{2\,1}(r)\,\left[\sqrt{1/3}\,Y_{1\,0}(\theta,\varphi)\,\chi_+ + \sqrt{2/3}\,Y_{1\,1}(\theta,\varphi)\,\chi_-\right].\]
- What values would a measurement of yield, and with what probabilities?
- Same for .
- Same for .
- Same for .
- Same for .
- Same for .
- What is the probability density for finding the electron at , , ?
- What is the probability density for finding the electron in the spin up state (with respect to the -axis) at radius ?

- In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by
- In the spin singlet (i.e., spin zero) state.
- In the spin triplet (i.e., spin one) state.

- Consider two electrons in a spin singlet (i.e., spin zero) state.
- 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 ?
- 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 ?
- 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?

### Contributors

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