Quantum Physics II, Test Formula Sheet | Massachusetts Institute of Technology - Edubirdie

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Massachusetts Institute of Technology
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8.05 | Quantum Physics II
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2021
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Formula Sheet • Conservation of probability ∂ ∂ ρ(x, t) + J(x, t) = 0 ∂t ∂x ~ ∂ ∗ ∗ ∂ 2 ρ(x, t) = |ψ(x, t)| ; J(x, t) = ψ ψ−ψ ψ 2im ∂x ∂x • Variational principle: dx ψ ∗ (x)Hψ(x) Egs ≤ R , for all ψ(x) dxψ ∗ (x)ψ(x) R • Spin-1/2 particle: Stern-Gerlach : µB = ~, H = −~µ · B e~ , 2me ~µ = g ~µe = −2 µB e~ 1 ~ ~ S = γS 2m ~ ~ S , ~ 1 0 In the basis |1i ≡ |z; +i = |+i = , |2i ≡ |z; −i = |−i = 0 1 ~ Si = σi 2 σx = 0 1 1 0 [σi , σj ] = 2iǫijk σk → σi σj = δij I + iǫijk σk ; σy = 0 −i i 0 ; σz = Si , Sj = i~ ǫijk Sk 1 0 0 −1 (ǫ123 = +1) → (~σ · ~a)(~σ · ~b) = ~a · ~b I + i~σ · (~a × ~b) eiMθ = 1 cos θ + iM sin θ , if M2 = 1 ~a exp i~a · ~σ = 1 cos a + i~σ · sin a , a a = |a| ~ exp(iθσ3 ) σ1 exp(−iθσ3 ) = σ1 cos(2θ) − σ2 sin(2θ) exp(iθσ3 ) σ2 exp(−iθσ3 ) = σ2 cos(2θ) + σ1 sin(2θ) . ~ = nx Sx + ny Sy + nz Sz = ~ ~n · ~σ . S~n = ~n · S 2 (nx , ny , nz ) = (sin θ cos φ, sin θ sin φ, cos θ) , |~n; +i = cos(θ/2)|+i + S~n |~n; ±i = ± sin(θ/2) exp(iφ)|−i |~n; −i = − sin(θ/2) exp(−iφ)|+i + cos(θ/2)|−i ~ |n; ~ ±i 2 • Bras and kets: For an operator Ω and a vector v, we write |Ωvi ≡ Ω|vi Adjoint: hu|Ω† vi = hΩu|vi|α1 v1 + α2 v ∗ ∗ 2 i = α1 |v1 i + α2 |v2 i ←→ hα1 v1 + α2 v2 | = α1 hv1 | + α2 hv2 | • Complete orthonormal basis |ii hi|ji = δij , 1= X |iihi| X Ωij |iihj | i Ωij = hi|Ω|ji ↔ Ω = i,j † hi|Ω |ji = hj|Ω|ii∗ Ω hermitian: Ω† = Ω, • Matrix M is normal ([M, M † ] = 0) ←→ U unitary: U † = U −1 unitarily diagonalizable. ˜ • Position and momentum representations: ψ(x) = hx|ψi ; ψ(p) = hp|ψi ; Z x̂|xi = x|xi , hx|yi = δ(x − y) , 1 = dx |xihx| , x̂† = x̂ Z p̂|pi = p|pi , hq|pi = δ(q − p) , 1 = dp |pihp| , p̂† = p̂ Z Z 1 ipx 1 ipx hx|pi = √ exp ; ψ̃(p) = dxhp|xihx|ψi = √ dx exp − ψ (x) ~ ~ 2π~ 2π~ ~ d n d n ~ n n hx|p̂ |ψi = ψ(x) ; hp|x̂ |ψi = i~ ψ̃(p) ; [p̂, f (x̂)] = f ′ (x̂) i dx dp i Z ∞ 1 exp(ik x)dx = δ(k) 2π −∞ • Generalized uncertainty principle (∆A)2 ≡ h(A − hAi)2 i = hA2 i − hAi2 2 1 (∆A) (∆B) ≥ hΨ| [A, B]|Ψi 2i 2 2 ∆x ∆p ≥ ~ 2 1 x2 ∆ ~ ∆x = √ and ∆p = √ for a gaussian wavefuntion ψ ∼ exp − 2 ∆2 2 2∆ r Z +∞ π dx exp −ax2 = a −∞ Time independent operator Q : ∆H∆t ≥ ~ , 2 ∆t ≡ d i hQi = h [H, Q]i dt ~ ∆Q dhQi dt • Commutator identities [A, BC] = [A, B]C + B[A, C] , 1 1 eA Be−A = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . . , 2 3! eA Be−A = B + [A, B] , if [[A, B], A] = 0 , [ B , eA ] = [ B , A ]eA , if [[A, B], A] = 0 1 1 eA+B = eA eB e− 2 [A,B] = eB eA e 2 [A,B] , if [A, B] commutes with A and with B • Harmonic Oscillator 1 2 1 1 2 2 Ĥ = p̂ + mω x̂ = ~ω N̂ + , N̂ = â† â 2m 2 2 r r ip̂ mω mω ip̂ † , â = â = x̂ + x̂ − , 2~ mω 2~ mω r r ~ mω~ † † x̂ = (â + â ) , p̂ = i (â − â) , 2mω 2 [x̂, p̂] = i~ , [â, â† ] = 1 , [N̂ , â ] = −â , ˆ , â† ] = â† . [N 1 |ni = √ (a† )n |0i n! Ĥ|ni = En |ni = ~ω n + √ 1 |ni , N̂ |ni = n|ni , hm|ni = δmn 2 √ n + 1|n + 1i , â|ni = n|n − 1i . mω 1/4 mω ψ0 (x) = hx|0i = exp − x2 . π~ 2~ â† |ni =

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