(a) (10 pts) Assume the potential can be written as the sum of threeone-dimensional potentials,V(x, y, z) =Vx(x) +Vy(y) +Vz(z).(4)Also assume that the eigenfunctions can be represented as theproductof three one-dimensional functions,ψjk‘(x, y, z) =X(x)Y(y)Z(z).(5)2
Plug the potential of Eq. 4 and the eigenfunction of Eq. 5 into theeigenvector equation for the Hamiltonian (the time-independentSchr¨odinger equation),-¯h22m∂2∂x2+∂2∂y2+∂2∂z2!ψjk‘+V(x, y, z)ψjk‘=Ejk‘ψjk‘.(6)Show that Eq. 6 reduces to three separateone-dimensionalTISE’s,one for each of the three functionsX(x),Y(y), andZ(z), and findan expression forEjk‘in terms of the eigenvalues of the 1D TISE’s.Hint:Use the following steps, which are standard when doingseparation of variables: Divide the TISE byXY Z.You shouldhave four terms: one that depends only onx, one that dependsonly ony, one that depends only onz, and one that is independentofx, y, z.Argue that each of the four terms must therefore beindependent ofx, y, z, that is equal to a constant.(b) (10 pts) Use the results above to show that the 3D ISW problemcan be reduced to three separate 1D ISW problems. Use the 1DISW solution (Townsend problem 6.13, no need to re-derive theseresults) to show that the eigenvalues of the 3D ISW Hamiltonianare of the formEjk‘=C1j2+C2k2+C3‘2(7)wherej,k, and‘are any positive integers and find the constantsC1,C2,C3.Also find the properly normalized eigenfunctionsψjk‘(x, y, z).
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