Download Final Exam Formula Sheet - Quantum Mechanics and Atomic Physics 1 | PHYS 3220 and more Exams Quantum Mechanics in PDF only on Docsity! p. 1 of 4 Phys3220 Final exam , Fall 2008 Name ____________________ ON ALL PARTS, you will be graded on the quality of your explanations. Please be brief but clear! (A correct answer with no explanation or indication of how you got it will not be worth any points.) • IMPORTANT! There are 6 numbered problems on this exam (each with multiple parts), but we will drop your single worst numbered problem. (You can work on all 6 and we'll take the best 5, or you can simply choose to ignore any one numbered problem.) So, as soon as the exam begins, please briefly look over the entire exam. • Each numbered problem is worth the same total number of points. (Subparts will be scored based on their relative length/difficulty within the problem.) • You may use 3 sides of your own handwritten notes. On the next page we provide a crib sheet with what we think are useful formulas. • Calculators are allowed (but of course no phones, internet, "electronic crib sheets", etc!) • Each question extends to the back of the page (they're not that long, we're just giving you space to work) If you need more room, staple extra pages to the exam. BE SURE TO INDICATE CLEARLY that your work is continued, (and where) if you need more room than the problem page itself. p. 2 of 4 Formula Sheet for Final: Steve and Oliver's Crib sheet: The time-dependent Schrödinger Equation, (TDSE): € i∂Ψ( r ,t) ∂t = − 2 2m ∇2Ψ( r ,t) + V ( r )Ψ( r ,t). (In 1-D, just let € ∇2 → ∂ 2 ∂x 2 ) The time-independent Schrödinger Equation (TISE): € ˆ H u( r ) = E u( r ), or € − 2 2m ∇2u( r ) + V ( r )u( r ) = Eu( r ) (In 1-D, just let € r→ x, and ∇2 → ∂ 2 ∂x 2 ) The standard deviation of an operator O: € σ ˆ 0 = (ΔO) 2 = O − O( )2 = O2 − O 2 The momentum operator: € ˆ p x = i ∂ ∂x € 2 ≈1.414 Harmonic Oscillator: x and p in terms of raising and lowering operators: € ˆ x = 2mω (a+ + a−), ˆ p = i mω 2 (a+ − a−) Normalization for raising and lowering operators: In Dirac Notation: € a+ n = n +1 n +1 a− n = n n −1 In older notation: € a+un (x) = n +1 un+1(x) a−un (x) = nun−1(x) (where the u's are the usual energy eigenfunctions of the Harmonic Oscillator, with u0(x) the ground state)
