Quantum Mechanics: Substitution and Coulomb Potential in Schrodinger Equation

[spaghetti3451]

Homework Statement

(i) Briefly indicate how substitution of operators corresponding to dynamical variables in an eigenvalue equation leads to the Schrodinger equation [itex]\left( \frac{-ħ^{2}}{2m} ∇^{2} + V \right)ψ = Eψ.[/itex]

(ii) What is the Coulomb potential, V(r), of an electron, charge e, in a hydrogen atom at distance r from the nucleus?

(iii), (iv), (v) left out for the moment

Homework Equations

The Attempt at a Solution

(i) (T + V) = E : law of conservation of energy

Multiply by ψ to obtain an eigenvalue equation: (T + V)ψ = Eψ

Substitute operators [itex]\widehat{T}[/itex] and [itex]\widehat{V}[/itex] corresponding to the dynamical variables T and V in the eigenvalue equation: [itex]( \widehat{T} + \widehat{V} ) ψ = Eψ[/itex]

[itex]\widehat{T} = \frac{\widehat{p}^{2}}{2m} = \frac{(-iħ∇)^{2}}{2m} = \frac{-ħ^{2}}{2m} ∇^{2}[/itex]

[itex]\widehat{V} = V[/itex]

So, the eigenvalue equation becomes the Schrodinger equation [itex]\left( \frac{-ħ^{2}}{2m} ∇^{2} + V \right)ψ = Eψ[/itex].(ii) V(r) = [itex]\frac{-e^{2}}{4πε₀r}[/itex]

Any comments would be greatly appreciated.

 

[BruceW]

yep, your answers look good to me.