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# Exercises

1. Calculate the Clebsch-Gordon coefficients for adding spin one-half to spin one.

2. An electron in a hydrogen atom occupies the combined spin and position state whose spinor-wavefunction is

Here, are the eigenstates of corresponding to the eigenvalues , respectively, and , , are conventional spherical coordinates.
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 ?
[61]

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

where is the vector connecting the two particles, , 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.] [53]

4. 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? Here, are the eigenstates of corresponding to the eigenvalues, , respectively, for the electron in question. [53]

Next: Time-Independent Perturbation Theory Up: Addition of Angular Momentum Previous: Calculation of Clebsch-Gordon Coefficients
Richard Fitzpatrick 2016-01-22