For an atom with one electron and nuclear charge of Z, the Hamiltonian is:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] H=-~\frac{\nabla^{2}}{2}~- ~\frac{Z}{r}~ [/tex]

1) show that the wavefunction:

[tex]\Psi_{1s}=Ne^{-Zr}[/tex]

is an eigenfunction of the Hamiltonian

2) find the corresponding energy

3) find N, the normalisation constant

In spherical polar coordinates:

[tex]\nabla^{2}\Psi_{1s}=~\frac{1}{r^{2}}~(~\frac{d}{dr}~[r^{2}~\frac{d\Psi_{1s}}{dr}])[/tex]

by applying H to the wavefunction, i think i've shown that it's an eigenfunction:

1)

[tex]H\Psi_{1s}=~\frac{-Z^{2}r^{2}}{2}~Ne^{-zr} - ~\frac{Z}{r}~[/tex]

[tex]H\Psi=n\Psi [/tex]

where n is the eigenvalue, and the bit on the end:

[tex]- ~\frac{Z}{r}~[/tex]

doesn't matter right, i've still shown it's an eigenvalue?

2)

to find the corresponding energy, don't I need to know N first?

3)

to find N, am I right in thinking:

[tex] N^{2} \int \Psi* \Psi dx = 1[/tex]

[tex]N^{2} \int e^{-2zr} dx = 1[/tex]

[tex]~\frac{-N^{2}}{2z}~e^{-2zr}=1[/tex]

[tex]N=\sqrt{~\frac{-2}{e^{-2zr}}~}[/tex]

but i think i must have gone wrong somewhere, i mean that doesn't look right. Once i've found N, how do i find the corresponding energy do i just plug N into:

[tex]H\Psi_{1s}=~\frac{-Z^{2}r^{2}}{2}~Ne^{-zr} - ~\frac{Z}{r}~[/tex]

ie, [tex]H\Psi_{1s}[/tex] = the corresponding energy?

i'd appreciate any help, thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Hamiltonian Operator

**Physics Forums | Science Articles, Homework Help, Discussion**