A particle of mass moves in the harmonic potential .

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

A particle of mass mm moves in the harmonic potential V(r⃗)=12m(ωx2x^2+ωy2y^2+ωz2z^2)V(\vec r) = \tfrac12 m(\omega_x^2\hat x^2+\omega_y^2\hat y^2+\omega_z^2\hat z^2). ((a)) Suppose ωx,ωy,ωz\omega_x,\omega_y,\omega_z are all different. Using [L^z,x^]=iℏy^[\hat L_z,\hat x]=i\hbar\hat y, [L^z,y^]=−iℏx^[\hat L_z,\hat y]=-i\hbar\hat x, [L^z,p^x]=iℏp^y[\hat L_z,\hat p_x]=i\hbar\hat p_y, [L^z,p^y]=−iℏp^x[\hat L_z,\hat p_y]=-i\hbar\hat p_x, and [L^z,z^]=[L^z,p^z]=0[\hat L_z,\hat z]=[\hat L_z,\hat p_z]=0, compute [H^,L^z][\hat H,\hat L_z] explicitly. Show it is non-zero in general, and explain why this means no single radial equation of the usual type can be written for this potential. ((b)) Now set ωx=ωy=ωz=ω\omega_x=\omega_y=\omega_z=\omega, an isotropic and hence central potential. Show [H^,L⃗^]=0[\hat H,\hat{\vec L}]=0, state the resulting decomposition of an energy eigenspace, and obtain the equation for u(r)=rR(r)u(r)=rR(r) together with the boundary condition at r=0r=0, giving Veff(r)V_{\rm eff}(r) explicitly.

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