# Question on quantum mechanics

1. Apr 22, 2012

### spaghetti3451

1. The problem statement, all variables and given/known data

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

(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

2. Relevant equations

3. 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 $\widehat{T}$ and $\widehat{V}$ corresponding to the dynamical variables T and V in the eigenvalue equation: $( \widehat{T} + \widehat{V} ) ψ = Eψ$

$\widehat{T} = \frac{\widehat{p}^{2}}{2m} = \frac{(-iħ∇)^{2}}{2m} = \frac{-ħ^{2}}{2m} ∇^{2}$

$\widehat{V} = V$

So, the eigenvalue equation becomes the Schrodinger equation $\left( \frac{-ħ^{2}}{2m} ∇^{2} + V \right)ψ = Eψ$.

(ii) V(r) = $\frac{-e^{2}}{4πε₀r}$

Any comments would be greatly appreciated.

2. Apr 23, 2012