Q. Consider a 3-d isotropic harmonic oscillator. Choose any value for the frequency. Find its energy eigenvalues, and plot the radial wave functions for the first 3 eigenvalues, in a graph that also shows the potential. Indicate the degeneracies if any. I used MATLAB for this plot. Note: After finding normalisation constant and radial wavefunction, I have scaled it by a factor of ½ for the plot, to get a clearer shape. I have plotted each radial wavefunction near the energy level it corresponds to, by adding E(n) do it. This is similar to what JJ Griffiths does in the textbook, to understand the shape of the wavefunction. REFERENCES: https://nukephysik101.wordpress.com/2018/01/19/3d-harmonic-oscillator/ An Introduction to Quantum Mechanics by JJ Griffiths CODE: format long h = 6.63e-34; p=pi; hbar=h/(2*p); m=hbar*hbar; w=(1/m)^(1/2); alpha=(hbar/(m*w))^(1/2); r=0:0.01:4; hold on; f1=integralConstant(0,0,alpha).*radialFunction(0,0,r,alpha).*(1/2)+calculateEnergy(0,0,hbar,w,r); f2=integralConstant(0,1,alpha).*radialFunction(0,1,r,alpha).*(1/2)+calculateEnergy(0,1,hbar,w,r); f3=integralConstant(1,0,alpha).*radialFunction(1,0,r,alpha).*(1/2)+calculateEnergy(1,0,hbar,w,r); f4=integralConstant(0,2,alpha).*radialFunction(0,2,r,alpha).*(1/2)+calculateEnergy(0,2,hbar,w,r); potential=calculatePotential(r,m,w); a1=plot (r,f1) E1=plot(r,calculateEnergy(0,0,hbar,w,r)) a2=plot(r,f2) E2=plot(r,calculateEnergy(0,1,hbar,w,r)) a3=plot(r,f3) E3=plot(r,calculateEnergy(1,0,hbar,w,r)) a4=plot(r,f4) E4=plot(r,calculateEnergy(0,2,hbar,w,r)) v1=plot(r,potential) M1="Radial wavefunction k=0 l=0"; M2="Radial wavefunction k=0 l=1"; M3="Radial wavefunction k=1 l=0"; M4="Radial wavefunction k=0 l=2 DEGENERATE with k=1 l=0"; M5="Energy k=0 l=0"; M6="Energy k=0 l=1"; M7="Energy k=1 l=0"; M8="Energy k=0 l=2 (same as Energy k=1 l=0)"; M9="Potential" xlabel("Radial distance r") ylabel("Energy") legend([a1,a2,a3,a4,E1,E2,E3,E4,v1], [M1, M2,M3,M4,M5,M6,M7,M8,M9]); hold off; function V=calculatePotential(r,m,w) V=(1/2).*m.*w.*w.*r.*r; end function E=calculateEnergy(k,l,hbar,w,r) E=hbar*w*(2*k+l+(3/2))+0.*r; end function R=radialFunction(k,l,r,alpha) R=(r.^l).*exp((-r.*r)/(2*alpha*alpha)).*laguerreL(k,(l+(1/2)), (r.*r)/(alpha*alpha)); end function N=integralConstant(k,l,alpha) temp1=integral((@(x)radialFunction(k,l,x,alpha).*radialFunction(k,l,x,alpha).*x.*x),0,Inf); temp2=1/temp1; N=(temp2)^(1/2); end