6.P: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
- 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 ψ=R21(r)[√1/3Y10(θ,φ)χ++√2/3Y11(θ,φ)χ−].
- What values would a measurement of L2 yield, and with what probabilities?
- Same for S2 .
- Same for J2 .
- Same for r , φ ?
- 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 where σ σ V(x) with respect to the overall spin 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 z -component of the spin of the other electron yields Sy=ℏ/2 , what is the probability that a measurement of the Sx=−ℏ/2 ?
- 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)