FormulaSheetFinalExam

# FormulaSheetFinalExam - 1 of 5 Formula Sheet for Final...

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1 of 5 Formula Sheet for Final Exam (These formulas will be given.) The classical wave equation: 22 2 f(x,t) 1 xv t ∂∂ = The time-dependent Schrödinger Equation: 2 iV t2 m x ∂Ψ ∂ Ψ =− + Ψ = = ( x ) The standard deviation () 2 2 2 2 xx σ= = = x Momentum operator: x ˆp ix = = 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= * ** i1 ˆ J(x,t) Im Re p 2m x x m x m ⎛⎞ ≡Ψ− Ψ = Ψ = Ψ Ψ ⎜⎟ ⎝⎠ == * Momentum space wavefunction : 1 2 1 2 (x,t) dp (p,t)exp(i p x / ) (p,t) dx (x,t)exp( ip x / ) π π Ψ= Φ Φ= Ψ = = = = Position-momentum commutator: x ˆ ˆ [x, p ] i = = dQ i ˆ ˆ [H, Q] dt = = (assuming Q operator is independent of time) Heisenberg Uncertainty Principle: AB 11 ˆˆ [A, B] [A, B] 2i 2 σσ ≥ = ij ˆ [L ,L ] i L = = k 5/2/2008 M.Dubson, PHYS3220 U.Colorado at Boulder

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2 of 5 Radial TISE for H-atom: () 22 2 eff eff du Vu E u,V V ( r ) 1 2m dr 2mr −+ = = + == AA + i Y(, ) ( s in )e φ θφ = θ A A ˆ L ,m ( 1 )m (m1 ) ,m1 ± =+ −± ± A A Spin ½ matrices: xy z 01 0 i 1 0 ˆˆ ˆ SS S 10 i 0 0 1 22 2 ⎛⎞ ⎛ ⎞ == = ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ / k d k **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 computation of expectation values and standard deviation, given a wavefunction Classical wave equation,
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FormulaSheetFinalExam - 1 of 5 Formula Sheet for Final...

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