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# 10.E: Addition of Angular Momentum (Exercises)

1. An electron in a hydrogen atom occupies the combined spin and position state $R_{2,1}(r)\,\left[\sqrt{1/3}\,Y_{1,0}(\theta,\phi)\,\chi_+ + \sqrt{2/3}\,Y_{1,1}(\theta,\phi)\,\chi_-\right].$

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 $$r$$, $$\theta$$, $$\phi$$?

8. What is the probability density for finding the electron in the spin up state (with respect to the $$z$$-axis) at radius $$r$$?

2. In a low energy neutron-proton system (with zero orbital angular momentum), the potential energy is given by $V(r) = V_1(r) + V_2(r)\left[3\,\frac{(\bsigma_1\cdot{\bf r})\,(\bsigma_2\cdot {\bf r})}{r^2} -\bsigma_1\cdot\bsigma_2\right] + V_3(r)\,\bsigma_1\cdot\bsigma_2,$ where $$\bsigma_1$$ denotes the vector of the Pauli matrices of the neutron, and $$\bsigma_2$$ denotes the vector of the Pauli matrices of the proton. Calculate the potential energy for the neutron-proton system:

1. In the spin singlet state.

2. In the spin triplet state.

3. Consider two electrons in a spin singlet 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 $$S_x=-\hbar/2$$?

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 state?

# Contributors

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

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