Lecture 14

# Lecture 14 - Hydrogen atom Classical mechanics in a...

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Unformatted text preview: Hydrogen atom Classical mechanics in a Coulomb (gravitational potential) θ ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin sin 2 2 2 2 2 2 2 ( ); ( ) 2 2 2 2 r r r r r eff eff p pr p p p p p H m r m m r m r m m r mr p p L L V r V r m r mr m r mr τ θ θ α α α α α α =- = +- =- + =- + =- + = + = - + Motion in the central potential in classical physics is characterized by conserving angular momentum sin L pr θ = Due to this conservation law momentum of the particle changes when its radial coordinate changes. This can be interpreted as presence of some fictitious force affecting radial motion. Formally it can be demonstrated by separating kinetic energy into radial and azimuthal components: This potential is responsible for orbital motion of planets provided their total energy is negative. Schrodinger equation in central potentials 2 2 2 2 2 2 2 1 1 1 1 sin sin sin r r r r r θ θ θ θ θ ϕ ∂ ∂ ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ ∂ ∂ It is convenient to study properties of particles in the central potential presenting Schrödinger equation 2 2 ( ) ( , , ) ( , , ) 2 H V r r E r m ψ θ ϕ ψ θ ϕ = - ∇ + = h In spherical coordinates. Laplacian in spherical coordinates have the form of: Which can be separated in the radial and angular parts. The angular part 2 2 2 2 2 1 1 sin sin sin L θ θ θ θ θ ϕ ∂ ∂ ∂ + = - ∂ ∂ ∂ h Comparing the angular part with the operator of the square of the angular momentum in the spherical coordinates 2 2 2 2 2 1 1 ˆ sin sin sin L θ θ θ θ θ ϕ ∂ ∂ ∂ = - + ∂ ∂ ∂ h We can present the latter as Separation of variables 2 2 2 2 2 ( ) ( ) ( ) 2 2 ( 1) d dR r V r R r ER r mr dr dr mr l l - + + = + h h Hamiltonian can now be presented as 2 2 2 2 2 ( ) 2 2 L H r V r mr r r mr ∂ ∂ = - + + ∂ ∂ h This Hamiltonian commutes with the operator of the angular momentum, thus the eigenfunctions of the former must also be eigenfunctions of the latter. We ca, therefore, present these eigenfunctions as ( 29 ( 29 ( , , ) , lm r R r Y ψ θ ϕ θ ϕ = Where the last factor is the spherical harmonics, which represent eigenstates of the angular momentum in the coordinate representation. The first factor is some arbitrary function of the radial coordinate, which does not affect the angular dependence of the wave function. Substituting this expression into the Shrodinger equation and using that 2 2 ( , ) ( 1) ( , ) lm lm L Y l l Y θ ϕ θ ϕ = + h We can obtain an equation for the radial function: Radial equation 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) 0; 1 ( ) ( ) ( ) ( ) ; ( ); ( ) 2 ( ) ( 2 ) ( 1) ( 1) ( ) ( ) ( 1) d dR m u r r r V r E R r R dr dr r r dR du r u r dR du r d dR d u r r r u r r r dr r dr r dr dr dr dr dr d u r m r r V r E u r m dr r l l l l...
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Lecture 14 - Hydrogen atom Classical mechanics in a...

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