Quantum Physics II, Formula Sheet

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: Egs ≤ • Spin-1/2 particle: R dx ψ ∗ (x)Hψ(x) R ≡ hH iψ dxψ ∗ (x)ψ(x) Stern-Gerlach : H = −µ · B , µB = e~ , 2me for all ψ(x) 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 + iσ · (a × b) eiMθ = 1 cos θ + i M sin θ , if M2 = 1 a
exp i a · σ = 1 cos a + iσ · 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) . Sn = n · S = nx Sx + ny Sy + nz Sz = (nx , ny , nz ) = (sin θ cos φ, sin θ sin φ, cos θ) , |n; +i = cos( 12 θ)|+i + ~ n·σ . 2 Sn |n; ±i = ± ~ |n; ±i 2 sin( 12 θ) exp(iφ)|−i |n; −i = − sin( 12 θ) exp(−iφ)|+i + cos( 12 θ)|−i hn′ ; +|n; +i = cos( 21 γ) , γ is the angle between n and n′  iαS  ~ n Rotation operator: Rα (n) ≡ exp − hSin = n , 2 ~ 1 • Linear algebra Matrix representation of T in the basis (v1 , . . . , vn ) : T vj = X Tij vi i basis change: uk = X T ({u}) = A−1 T ({v})A Ajk vj , j Schwarz: |hu, vi| ≤ |u| |v| Adjoint: hu, T vi = hT † u, vi , (T † )† = T • 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 v2 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 hq|pi = δ(q − p) , 1 = dp |pihp| , p̂† = p̂ Z Z ipx
1 ipx
1 exp ; ψ̃(p) = dxhp|xihx|ψi = √ dx exp − ψ (x) hx|pi = √ ~ ~ 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(ikx)dx = δ(k) 2π −∞ p̂|pi = p|pi , 2 • Generalized uncertainty principle ∆A ≡ |(A − hAi1)Ψ (∆A)2 = hA2 i − hAi2 → ∆A ∆B ≥ hΨ| 1 [A, B]|Ψi 2i ∆x ∆p ≥ ~ 2  1 x2  ∆ ~ ∆x = √ and ∆p = √ for 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 = eadA B = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . . , 2 3! eA Be−A = B + [A, B] , if [A, [A, B] ] = 0 , [ B , eA ] = [ B , A ]eA , if [A, [A, B] ] = 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 2m 2 2 r r     mω ip̂ mω ip̂ † â = x̂ + , â = x̂ − , 2~ mω 2~ mω r r ~ mω~ † † x̂ = (â + â ) , p̂ = i (â − â) , 2mω 2 [x̂, p̂] = i~ , [â, ↠] = 1 , [N̂ , â ] = −â , ˆ , ↠] = ↠. [N 1 |ni = √ (a† )n |0i n! Ĥ|ni = En |ni = ~ω n + 1
|ni , N̂ |ni = n|ni , hm|ni = δmn 2 3 √ √ n + 1|n + 1i , â|ni = n|n − 1i .  mω 1/4  mω  ψ0 (x) = hx|0i = exp − x2 . π~ 2~ ↠|ni = p̂ sin ωt mω pH (t) = p̂ cos ωt − mω x̂ sin ωt xH (t) = x̂ cos ωt + • Coherent states and squeezed states i Tx0 ≡ e− ~ p̂ x0 , Tx0 |xi = |x + x0 i i 2 1 x0 |x̃0 i = e− 4 d2 e † −α∗ a |x̃0 i ≡ Tx0 |0i = e− ~ p̂x0 |0i , x √ 0 a† 2d |αi ≡ D(α)|0i = eαa |0i , † −α∗ a 1 |αi = eαa |0i , hx|x̃0 i = ψ0 (x − x0 ) ,   D(α) ≡ exp αa† − α∗ a , ~ mω hx̂i hp̂i d α= √ + i√ ∈C 2d 2~ d2 = † 2 |0i = e− 2 |α| eα a |0i , â|αi = α|αi , |α, ti = e−iωt/2 |e−iωt αi   γ |0γ i = S(γ)|0i , S(γ) = exp − (a† a† − aa) , γ ∈ R 2   1 1 † † |0γ i = √ exp − tanh γ a a |0i 2 cosh γ S † (γ) aS(γ) = cosh γ a − sinh γ a† , D † (α) aD(α) = a + α |α, γi ≡ D(α)S(γ)|0i • Time evolution |Ψ, ti = U(t, 0)|Ψ, 0i , U unitary U(t, t) = 1 , U(t2 , t1 )U(t1 , t0 ) = U(t2 , t0 ) , U(t1 , t2 ) = U † (t2 , t1 ) i~ d d ˆ ˆ |Ψ, ti = H(t)|Ψ, ti ↔ i~ U(t, t0 ) = H(t)U(t, t0 ) dt dt h i i X i ˆ Time independent H: U(t, t0 ) = exp − Ĥ(t − t0 ) = e− ~ En (t−t0 ) |nihn| ~ n hAi = hΨ, t|AS |Ψ, ti = hΨ, 0|AH (t)|Ψ, 0i [AS , BS ] = CS i~ → → AH (t) = U † (t, 0)AS U(t, 0) [AH (t) , BH (t) ] = CH (t) d ÂH (t) = [AˆH (t), ĤH (t)] , for AS time-independent dt 4 • Two state systems H = h0 1 + h · σ = h0 1 + h n · σ , h = |h| Eigenstates: |n; ±i , E± = h0 ± h . H = −γ S · B → spin vector ~n precesses with Larmor frequency ω = −γB
NMR magnetic field B = B0 z + B1 cos ωt x − sin ωt y  i i  |Ψ(t)i = exp ωtŜz exp γBR · S t |Ψ(0)i ~ ~ ω
BR = B0 1 − z + B1 x ω0 • Orbital angular momentum operators a · b ≡ ai bi , (a × b)i ≡ ǫijk aj bk , ǫijk ǫipq = δjp δkq − δjq δkp , a2 ≡ a · a . ǫijk ǫijq = 2δkq . L̂i = ǫijk x̂j p̂k ⇐⇒ L = r × p = −p × r Vector u under rotation: ˆ i , ûj ] = i~ ǫijk ûk [L Scalar S under rotation: [ L̂i , S ] = 0 u, v vectors under rotations  L̂i , L̂j  → 2 ∂ 1 ∂2 + 2 ∇ = 2+ ∂r r ∂r r 2 L̂ = −~ L̂z = 1 Y0,0 (θ, φ) = √ ; 4π ~ ∂ ; i ∂φ 2   L × u + u × L = 2i~ u u · v is a scalar, u × v is a vector = i~ ǫijk L̂k ⇐⇒ L × L = i~ L , 2 • Spherical Harmonics =⇒ [ L̂i , L2 ] = 0 . ∂2 1 ∂2 ∂ + cot θ + ∂θ2 ∂θ sin2 θ ∂φ2   ∂2 ∂ 1 ∂2 + cot θ + ∂θ2 ∂θ sin2 θ ∂φ2   ∂ ∂ ±iφ L̂± = ~e ± + i cot θ ∂θ ∂φ Yℓ,m(θ, φ) ≡ hθ, φ|ℓ, mi r 3 Y1,±1 (θ, φ) = ∓ sin θ exp(±iφ) ; 8π 5 Y1,0 (θ, φ) = r 3 cos θ 4π • Algebra of angular momentum operators J (orbital or spin, or sum) [Ji , Jj ] = i~ ǫijk Jk ⇐⇒ J × J = i~ J ; J2 |jmi = ~2 j(j + 1)|jmi ; J± = Jx ± iJy , Jz |jmi = ~m|jmi , m = −j, . . . , j . (J± )† = J∓ [Jz , J± ] = ±~ J± , [ J2 , Ji ] = 0 1 1 Jx = (J+ + J− ) , Jy = (J+ − J− ) 2 2i [J 2 , J± ] = 0 ; [J+ , J− ] = 2~ Jz J2 = J+ J− + Jz2 − ~Jz = J− J+ + Jz2 + ~Jz p J± |jmi = ~ j(j + 1) − m(m ± 1) |j, m ± 1i • Angular momentum in the two-dimensional oscillator 1 1 † ]=1 âL = √ (âx + iây ) , âR = √ (âx − iây ) , [âL , â†L ] = [âR , âR 2 2 J+ = ~ â†R âL , |j, mi : J− = ~ â†L âR , j = 12 (NR + NL ) , ˆR − N̂L ) Jz = 12 ~(N m = 21 (NR − NL ) H = 0 ⊕ 21 ⊕ 1 ⊕ 32 ⊕ . . . • Radial equation  − ~2 ℓ(ℓ + 1)  ~2 d 2 + V (r) + uνℓ (r) = Eνℓ uνℓ (r) (bound states) 2m dr 2 2mr 2 uνℓ (r) ∼ r ℓ+1 , as r → 0 . • Hydrogen atom En = − e2 1 , 2a0 n2 ψn,ℓ,m (~x) = unℓ (r) Yℓ,m(θ, φ) r n = 1, 2, . . . , ℓ = 0, 1, . . . , n − 1 , m = −ℓ, . . . , ℓ ~2 e2 1 a0 = , α= ≃ , ~c ≃ 200 MeV–fm 2 me ~c 137 2r u1,0 (r) = 3/2 exp(−r/a0 ) a0   2r r u2,0 (r) = 1− exp(−r/2a0 ) (2a0 )3/2 2a0 1 1 r2 u2,1 (r) = √ exp(−r/2a0 ) 3 (2a0 )3/2 a0 6 • Addition of Angular Momentum J = J1 + J2 Uncoupled basis : |j1 j2 ; m1 m2 i CSCO : {J21 , J22 , J1z , J2z } Coupled basis : |j1 j2 ; jmi CSCO : {J21 , J22 , J2 , Jz } j1 ⊗ j2 = (j1 + j2 ) ⊕ (j1 + j2 − 1) ⊕ . . . ⊕ |j1 − j2 | |j1 j2 ; jmi = X m1 +m2 =m |j1 j2 ; m1 m2 i hj1 j2 ; m1 m2 |j1 j2 ; jmi {z } | Clebsch−Gordan coefficient 1 J1 · J2 = (J1+ J2− + J1− J2+ ) + J1z J2z 2 Combining two spin 1/2 : | 1, 1 i = | ↑↑i , 1 | 1, 0 i = √ (| ↑↓i + | ↓↑i) , 2 |1, −1i = | ↓↓i . 7 1 1 ⊗ = 1⊕0 2 2 1 |0, 0i = √ (| ↑↓i − | ↓↑i) 2