Consider a one-electron atom with a nucleus of charge Ze. Show that the wave functions (see Table 8-1) are the hydrogen atom wave functions with the Bohr radius replaced by (a₀)/(Z).
Spectroscopic Notation
Yn,l,m(r,θ,φ) = (1/ao)^(1/2) (1,0,0) √(1/π) e^(-r/ao)
1s
Y1,0,0(r,θ,φ) = (1/π)^(1/2) (2/ao)^(3/2) e^(-r/ao)
2s
Y2,0,0(r,θ,φ) = (1/π)^(1/2) (4/ao)^(3/2) (1 - r/2ao) e^(-r/2ao)
2p (m=0)
Y2,1,0(r,θ,φ) = (1/π)^(1/2) (4/ao)^(3/2) r/ao cos(θ) e^(-r/2ao)
2p (m=1)
Y2,1,1(r,θ,φ) = (1/π)^(1/2) (4/ao)^(3/2) r/ao sin(θ) e^(iφ) e^(-r/2ao)
2p (m=-1)
Y2,1,-1(r,θ,φ) = (1/π)^(1/2) (4/ao)^(3/2) r/ao sin(θ) e^(-iφ) e^(-r/2ao)
3s
Y3,0,0(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) (1 - 2r/3ao) e^(-r/3ao)
3p (m=0)
Y3,1,0(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r/3ao cos(θ) e^(-r/3ao)
3p (m=1)
Y3,1,1(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r/3ao sin(θ) e^(iφ) e^(-r/3ao)
3p (m=-1)
Y3,1,-1(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r/3ao sin(θ) e^(-iφ) e^(-r/3ao)
3d (m=0)
Y3,2,0(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r^2/3ao (3cos^2(θ) - 1) e^(-r/3ao)
3d (m=1)
Y3,2,1(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r^2/3ao sin(θ) cos(θ) e^(iφ) e^(-r/3ao)
3d (m=-1)
Y3,2,-1(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r^2/3ao sin(θ) cos(θ) e^(-iφ) e^(-r/3ao)
3d (m=2)
Y3,2,2(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r^2/3ao sin^2(θ) e^(2iφ) e^(-r/3ao)
3d (m=-2)
Y3,2,-2(r,θ,φ) = (1/π)^(1/2) (9/ao)^(3/2) r^2/3ao sin^2(θ) e^(-2iφ) e^(-r/3ao)