Science Homework Help

For a spherically symmetric state of a hydrogen atom, the Schrödinger equation in spherical coordinates is:

−ℏ22me(d2ψdr2+2rdψdr)−kee2rψ=Eψ$\frac{-{\hslash }^2}{2m_e}(\frac{d^2\psi }{d{r}^2}+\frac{2}{r}\frac{d\psi }{dr})-\frac{k_e{e}^2}{r}\psi =E\psi$

show that the 1s wave function for an electron in hydrogen,

ψ1s(r)=1πa3o√e−rao${\psi }_{1s}\left(r\right)=\frac{1}{\sqrt{\pi a_o^3}}{e}^{-\frac{r}{a_o}}$

satisfies the Schrödinger equation, with E = -13.6 eV, and r=ao (ground state).

Use: ao = 5.29×10-11m, ke = 8.988×109 Nm2/C2, ħ = 1.055×10-34 Js, me = 9.11×10-31 kg, e = 1.602×10-19 C, and 1 eV = 1.602×10-19 J.