SJ_MB_QMnotes

SJ_MB_QMnotes - 3D ∇ − = i p 1D dx d i p x − = Particle in a box Schrödinger equation ψ E x V dx d m = − 2 2 2 2 V(x = 0 by rearranging 2

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Plan of action: - Background; - “Postulates” of QM; - Application #1: Particle in a Box Application #2: Atomic orbitals; Hints leading to QM: - Discrete spectral lines; - Specific heats of solids (Law of Dulong and Petit, 3R); - Photoelectric effect; - Davisson-Gerner experiment (electron diffraction); Theoretical Responses: - Discrete spectral lines: Niels Bohr 2 2 4 2 2 1 2 6 . 13 n h me eV n E n π = = ; - De Broglie: “pilot waves” Attribute a wavelength to matter: p h = λ ; Classical Mechanics Quantum Mechanics 1) The “state” of a system { } i i p q , “wave function” ) ( x ψ 2) Evolution Equations i i q H p = ; i i p H q = H t i ˆ = 3) Observables and measurements Any mechanical variable we might be interested in is a function of p’s and q’s, e.g. 2 2 2 1 2 kq m p E + = i) dx x 2 ) ( probability that particle will be found between x and x+dx ii) “Expected” value of an observable, C ˆ , is given by C x C ˆ ) ( ˆ * = Question: What observables? In QM, observables are represented by “operators”
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Unformatted text preview: 3D: ∇ − = i p ; 1D: dx d i p x − = Particle in a box: Schrödinger equation: ψ E x V dx d m = + − ) ( 2 2 2 2 V(x) = 0 by rearranging: 2 2 2 2 = + mE dx d → x mE B x mE A 2 2 2 sin 2 cos + = Boundary conditions: ψ(0)= ψ(na)=0 → ditch cosine π n a mE = 2 2 → 2 2 2 2 2 ma n E n = Toy model of molecular binding: 2 2 2 " " 2 2 2 ma E atoms × = 2 2 2 ) ( 2 2 a m E molecule α × = − = − ⋅ = 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 ma ma a m E bond J J m kg s J E bond 17 18 20 30 68 2 10 31 2 2 2 34 10 10 9 10 10 10 4 36 ) 10 )( 10 9 ( 4 ) 10 6 ( − − − − − − − − ≈ × = = × ⋅ × ≈...
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This document was uploaded on 01/03/2012.

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SJ_MB_QMnotes - 3D ∇ − = i p 1D dx d i p x − = Particle in a box Schrödinger equation ψ E x V dx d m = − 2 2 2 2 V(x = 0 by rearranging 2

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