Quantum Mechanics Final Exam Equation Sheet, Cheat Sheet of Quantum Mechanics

Download Quantum Mechanics Final Exam Equation Sheet and more Cheat Sheet Quantum Mechanics in PDF only on Docsity! Quantum Mechanics Final Exam Equation Sheet 171.303 Quantum Mechanics I December 21st, 2017 Stern-Gerlach Experiments |ψ〉 = c+|+ z〉+ c−| − z〉 1 = 〈ψ|ψ〉 = |c+|2 + |c−|2 1 = |+ z〉〈+z|+ | − z〉〈−z| 〈+z| − z〉 = 〈−z|+ z〉 = 0 〈O〉 = 〈ψ|Ô|ψ〉 Representation |ψ〉 = c+|+ z〉+ c−| − z〉 → ( c+ c− ) and 〈ψ| → ( c∗+ c ∗ − ) e.g. | ± x〉 = 1√ 2 |+ z〉 ± 1√ 2 | − z〉 → ( 1√ 2 ± 1√ 2 ) | ± y〉 = 1√ 2 |+ z〉 ± i√ 2 | − z〉 → ( 1√ 2 ± i√ 2 ) For an operator Â→ A with Aij = 〈i|Â|j〉 Â|ψ〉 = |φ〉 =⇒ 〈i|φ〉 = 〈i|Â|ψ〉 = ∑ j 〈i|Â|j〉〈j|ψ〉 = ∑ j Aij〈j|ψ〉 1 = |+ z〉〈+z|+ | − z〉〈−z| = P̂+ + P̂− Change of Basis:( 〈+z′|ψ〉 〈−z′|ψ〉 ) = ( 〈+z′|+ z〉 〈+z′| − z〉 〈−z′|+ z〉 〈−z′| − z〉 )( 〈+z|ψ〉 〈−z|ψ〉 ) = S† ( 〈+z|ψ〉 〈−z|ψ〉 ) A′ = S†AS where A′ is in the new basis and A in the old basis 1 Rotation and Angular Momentum R̂(φn) = e−iŜnφ/~ Ŝ†n = Ŝn Spin-1 2 : Ŝn| ± n〉 = ±~2 | ± n〉 Ŝx → ~2 ( 0 1 1 0 ) , Ŝy → ~2 ( 0 −i i 0 ) , Ŝz → ~2 ( 1 0 0 −1 ) Ŝ→ ~ 2 σ (Pauli matrices) Any spin: Angular momentum algebra: [Ŝx, Ŝy] = i~Ŝz (and cyclic permutations) Raising/lowering operators Ŝ± = Ŝx ± iŜy [Ŝz, Ŝ±] = ±~Ŝ± Ŝ2|j,m〉 = s(s+ 1)~2|s,m〉 Ŝz|s,m〉 = m~|s,m〉 Ŝ±|s,m〉 = √ s(s+ 1)−m(m± 1)~|s,m± 1〉 s = 0, 1, 2, . . . (bosons) or s = 1/2, 3/2, 5/2, . . . (fermions) For fixed s, allowed states are m = −s,−s+ 1, . . . ,+s (2s+ 1 total) Uncertainty Principle: (∆A)2 = 〈A2〉 − 〈A〉2 If [Â, B̂] = iĈ, then ∆A∆B ≥ 1 2 |〈C〉| e.g. ∆Sx∆Sy ≥ ~2 |〈Sz〉| 2