FormulaSheetFinalExam

FormulaSheetFinalExam - 1 of 6 Formula Sheet for Final...

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1 of 6 Formula Sheet for Final Exam (These formulas will be given.) The classical wave equation: 2 2 2 2 2 f (x,t) 1 f (x,t) x v t = The time-dependent Schrödinger Equation: 2 2 2 i V(x) t 2m x Ψ Ψ = - + Ψ h h The standard deviation ( 29 ( 29 2 2 2 2 x x x x x σ = = - = - Momentum operator: x ˆp i x = h Fourier Transform formulae (Plancherel's Theorem) : 1 2 1 2 f (x) dk F(k)exp(ik x) F(k) dx f (x)exp( ik x) π π = = - A useful form of the delta function: 1 2 (x) exp(ik x)dk π δ = ( 29 * * * * i 1 ˆ J(x,t) Im Re p 2m x x m x m Ψ Ψ Ψ Ψ - Ψ = Ψ = Ψ Ψ h h Momentum space wavefunction : 1 2 1 2 (x,t) dp (p,t)exp(ip x / ) (p,t) dx (x,t)exp( ip x / ) π π Ψ = Φ Φ = Ψ - h h h h Position-momentum commutator: x ˆ ˆ [x, p ] i = h d Q i ˆ ˆ [H, Q] dt = h (assuming Q operator is independent of time) Heisenberg Uncertainty Principle: A B 1 1 ˆ ˆ ˆ ˆ [A, B] [A, B] 2i 2 σ σ = i j k ˆ ˆ ˆ [L ,L ] i L = h 2/27/2012 M.Dubson, PHYS3220 U.Colorado at Boulder
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2 of 6 Radial TISE for H-atom: ( 29 2 2 2 eff eff 2 2 d u V u Eu , V V(r) 1 2m d r 2mr - + = = + + h h l l i Y ( , ) (sin ) e φ θ φ = θ l l l l ˆ L ,m ( 1) m(m 1) ,m 1 = + - l l l l Spin ½ matrices: x y z 0 1 0 i 1 0 ˆ ˆ ˆ S S S 1 0 i 0 0 1 2 2 2 - = = = - h h h **End of Formulas** Exam 1 Review Topics: Ch 1 and Ch 2 in Griffiths, Homeworks 1 thru 5, and Lecture Notes up thru/including "Dirac Delta function" probability density and the wavefunction normalization of the wavefunction
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FormulaSheetFinalExam - 1 of 6 Formula Sheet for Final...

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