SJ_QMnotesSJ_QMnotes0001SJ_QMNotesp20001.pdfMB_QMnotes.pdfPlan 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 →22422126.13nhmeeVnEnπ−=−=; - De Broglie: “pilot waves” Attribute a wavelength to matter: ph=λ; Classical Mechanics Quantum Mechanics 1) The “state” of a system { }iipq, “wave function” )(xψ 2) Evolution Equations iiqHp∂∂−=; iipHq∂∂= ψψHtiˆ=∂∂ 3) Observables and measurements Any mechanical variable we might be interested in is a function of p’s and q’s, e.g. 22212kqmpE += i) ≡dxx2)(ψprobability that particle will be found between x and x+dx ii) “Expected” value of an observable, Cˆ, is given by CxCˆ)(ˆ*ψψψψ∫= Question: What observables? In QM, observables are represented by “operators” 3D: ∇−=ip; 1D: dxdipx−= Particle in a box: Schrödinger equation:ψψψExVdxdm=+− )(2222 V(x) = 0 by rearranging: 02222=+ψψmEdxd → xmEBxmEA222sin2cos+=ψ Boundary conditions: ψ(0)= ψ(na)=0 → ditch cosine πnamE=22 → 22222manEnπ= Toy model of molecular binding: 222""222maEatomsπ×= 222)(22amEmoleculeαπ×= −=−⋅=2222222222221222ααππαπmamaamEbond
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