ssp_2x - 0.3 Schrdinger equation p x = ih Eigenfunctions: (...

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0.3 Schrödinger equation pp x i x i =- ⇒ =- ∇ hh Eigenfunctions: ψψ () re ik r = 0 of eigenvalue pk = h Kinetic energy: classical m.: Em v p m kin == 1 22 2 2 q.m.: E kin p mm - 2 h operator of the kinetic energy Position operator: r ϕ = r Generalisation: Ur function of position ⇒= U rU r multiplication Total energy: EE E kin pot =+ Q.m: Hamiltonian (operator): HE kin H + h 2 2 m Eigenstates = states with sharp energy E H ψ = E -+= h 2 2 m r r Er () () : stationary states, only r not (,) rt e.g.: electronic states of atoms and molecules 21
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Time dependence of nonstationary states ψ (,) rt Operator of time derivative: ~ BB =→ = t i t Hermitian Eigenfunctions: i t ψϖ ϖϖ = ⇒= - ψϕ ϖ () re it , ϖ : eigenvalue scalar fields vibrating with ϖ Phys. meaning only r 2 2 = not time dependent! Eigenfunctions of B are stationary states. Given state Expansion in terms of eigenfunctions of B (continuous spectrum!) (, ) ar e d = - Fourier transform or spectrum 22
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Special case: state consisting of two eigenstates of B with ϖ 1 and ϖ 2 ψ ϖϖ (,) () rt a re it =+ -- 12 ψϖ ϖ c o s ( ) a a aa t 2 1 2 2 2 1 2 2 =++ - Spacial distribution vibrates harmonically with ϖ 1 - ϖ 2 , not stationary! B describes atom: Sharp stationary states i with frequency i : Cannot be physically observed as vibration.
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.

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ssp_2x - 0.3 Schrdinger equation p x = ih Eigenfunctions: (...

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