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Question on quantum mechanics

  1. Apr 22, 2012 #1
    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 [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

    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 [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.
     
  2. jcsd
  3. Apr 23, 2012 #2

    BruceW

    User Avatar
    Homework Helper

    yep, your answers look good to me.
     
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