Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 30

 1.6.(a) Derive the result, equation (1.269).(b) For d=1.00cm and neutrons of de Broglie wavelength λ=1.82Å, show that a 4π rotation is produced by ΔBd=149 Gauss ·cm. (μn=9.65×10−24 erg ·Gauss−1, m=1.67×10−27 kg.)

 1.7. Show that substituting Ψ(z)=χn(z) (cf. equation (1.170)) into equation (1.280) givesΨ(x)=Hn(x)e−x222nn!π.(1.341)Use the generating function for Hermite polynomialse−s2+2sx=∑nHn(x)snn!,withs=z*2.(1.342)

 1.8. Show that Ψ(z)=∑ncnznn!, where the cn are the expansion coefficients for Ψ(x) in the (orthonormal) basis defined by the one-dimensional harmonic oscillator energy eigenfunctions.

 1.9. Show that[Γ(J−),Γ(J+)]=−2Γ(J0),[Γ(J0),Γ(J+)]=+Γ(J+),[Γ(J0),Γ(J−)]=−Γ(J−).

 1.10. Show that:Γ(J2)≔12Γ(J−)Γ(J+)+Γ(J+)Γ(J−)+Γ(J0)2=j(j+1).