For a particle of mass in a central potential energy

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials

Problem

For a particle of mass mm in a central potential energy V(r)V(r), separation of the time-independent Schrödinger equation gives the radial equation−ℏ22m1r2ddr(r2dRdr)+[V(r)+ℏ2l(l+1)2mr2]R=ER.-\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\left[V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right]R = ER . ((a)) Substitute R(r)=u(r)/rR(r) = u(r)/r and show, term by term, that uu obeys −ℏ22md2udr2+Veff(r) u=Eu-\dfrac{\hbar^2}{2m}\dfrac{d^2u}{dr^2}+V_{\rm eff}(r)\,u = Eu, identifying VeffV_{\rm eff}. ((b)) An electron moves in the isotropic oscillator potential energy V(r)=12meω2r2V(r) = \tfrac12 m_e\omega^2r^2 with ℏω=0.500\hbar\omega = 0.500 eV. Write VeffV_{\rm eff} for l=1l = 1 and find the radius, in nm, at which it is a minimum. Use ℏ2/2me=0.0381\hbar^2/2m_e = 0.0381 eV nm2^2.

Related theory

Browse the problem library