Question

9. Write down the Hamiltonian operator in Cartesian coordinates for the hydrogen atom. Identify each term.
 

Answer 1

Hamiltonian operator is the operator for total energy of the system. The energy of hydrogen atom is the sum of potential energy (energy due to electrostatic attraction of electron and proton) and kinetic energy.

The kinetic energy operator is given as:

Ť = - (ħ² / 2m )Δ ²

ħ = h/2π; h = planks constant., m = mass of hydrogen atom, Δ² = laplacian operator

Δ² = (ƌ²/ƌx² + ƌ² /ƌy² + ƌ² /ƌz²)

Potential energy is given as:

V = - e²/4πϵ_or

e = charge on electron, r = distance of electron from nucleus, ϵ_o = permitivity

Hence Hamiltonian H is given as:

H =  - e²/4πϵ_or - (ħ² / 2m )Δ ²