Quantum Mechanics and the Hydrogen Atom

[a.11en]

Calculate the expectation value of the potential energy for an electron in a 1s orbital for a hydrogen atom





Ive determined the potential energy operator to be V=-e²/4∏ε₀r
and a wave function of

ψ= (1/4∏)^1/2

therefore i get
<V> = ∫∫∫ψ*Vψr²sin∅drd∅dphi
integrals from 0 to r, 0 to pi, 0 to 2pi


not sure where to go from here.

 

[a.11en]

Nevermind, i got it.

but if anyone is interested ill explain.

For a 1s orbital of a hydrogen the wavefunction is ψ=root(1/∏ao³ e^-r/ao)

this gives
∫∫∫ψVψr²sinθdrdθd∅
integrals are zero to 2pi, zero to pi, and zero to infinity.

then factor out any terms that are not a function of r,θ, or∅.
This gives several terms outside of the integral: ∫∫∫e^-2r/aorsinθdrdθd∅

then you can separate the integrals and evaluate. they were pretty easy to do.

the final answer was <v> = -e²/4∏ε_oa_o

 

[dextercioby]

You can use the virial theorem. The expectation value you're looking for is then half the ground state energy.