Schrödinger Equation in 3d
Consider a cubic “box” in which an electron of mass m is confined
outside the cube, we have U(x,y,z)=? and inside U(x,y,z)=0
hence electron has ?(x,y,z)=0 on all faces of the cube
?(x,y,z)=Asin(k1x)sin(k2y)sin(k3z)
?(L,y,z)=0 for all 0<y<L and 0<z<L => k1=n1?/L
similarly we need k2=n2 ?/L and k3=n3 ?/L
- E=(h2/2m)(k12+k22+k32) = (h2 ? 2/2mL2)(n12+n22+n32) =E1 (n12+n22+n32)
- where E1 is ground state energy of 1-d well