6.P: Exercises

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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
\[\psi = R_{2\,1}(r)\,\left[\sqrt{1/3}\,Y_{1\,0}(\theta,\varphi)\,\chi_+ + \sqrt{2/3}\,Y_{1\,1}(\theta,\varphi)\,\chi_-\right].\]
1. What values would a measurement of $ L^2$ yield, and with what probabilities?
2. Same for $ S^2$ .
3. Same for $ J^{\,2}$ .
4. Same for $ r$ , $ \varphi$ ?
5. What is the probability density for finding the electron in the spin up state (with respect to the $ r$ ?
In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by
$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 $ \sigma$ $ \sigma$ $ V({\bf x})$ with respect to the overall spin state.]
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 $ z$ -component of the spin of the other electron yields $ S_y=\hbar/2$ , what is the probability that a measurement of the $ S_x=-\hbar/2$ ?
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 (i.e., spin one) state?

Contributors

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

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