Quantum Mechanics Formulas by R.L. Griffith pdf - Web Education

Formula Sheet

• Conservation of probability ∂ ∂ ρ(x, t) + ∂t ∂xJ(x, t) = 0 ~ ρ(x, t) = |ψ(x, t)|2 ; J(x, t) = ψ ∂ 2im  ∗ _ψ ∂ ∂x − ψ ψ ∗ ∂x  • Variational principle: Egs ≤ R dx ψ∗_(x)Hψ(x) R H ψ for all ψ(x) dxψ∗_(x)ψ(x) ≡ h i • Spin-1/2 particle: e~ Stern-Gerlach : _{H = −µ · B ,} µ= g 2m 1 S= γS ~ e~ µB = S , µ 2me e =−2 µ B , ~ 1 0

In the basis |1i ≡ |z; +i = |+i = 

0 

, |2i ≡ |z; −i = |−i =  1  ~ Si = σi σx= 0 2  1 1 0  ; σy =  0 −i 1 0 ; σ = i 0  z  0 −1  [σi, σj] = 2iǫ σ →  S , S  = i~ ijk k i j ǫijkSk (ǫ123 = +1)

σiσj = δijI + iǫijkσk → (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 · a

a 

sin a , _{a = | |} exp(iθσ3) σ1 exp(−iθσ3) = σ1cos(2θ) − σ2sin(2θ)

exp(iθσ3) σ2 exp(−iθσ3) = σ2cos(2θ) + σ1sin(2θ) .

~ Sn = n · S = nxSx+ nySy+ nzSz = n

2 · σ . ~ (nx, ny, nz) = (sin θ cos φ, sin θ sin φ, cos θ) , Sn|n; ±i = ± n

2| ; ±i |n; +i = cos(1

2θ)|+i + sin(

1_{θ) exp(iφ)}

2 |−i

|n; −i = − sin(12θ) exp(−iφ)|+i + cos( 1_θ) 2 |−i



hn′; +|n; +i= cos(1γ) , γ is the angle between n and n′

2

~ hSin =

2n, Rotation operator: Rα(n) ≡ exp 

• Linear algebra

Matrix representation of T in the basis (v1, . . . , vn) : T vj =

X Tijvi i basis change: uk= X Ajkvj, T ({u}) = A−1T ( j {v})A 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

|α1v1+ α2v2i = α1|v1i + α2|v2i ←→ hα1v1+ α2v2| = α1∗hv1| + α∗2hv2|

• Complete orthonormal basis |ii

hi|ji = δij, 1 = X i |iihi| Ωij = hi|Ω|ji ↔ Ω = X Ωij |i j i,j ih | hi|Ω†_{|ji = hj|Ω|ii}∗

Ω hermitian: Ω† = Ω, U unitary: U† = U−1 • Matrix M is normal ([M, M†_{] = 0) ←→ unitarily diagonalizable.}

• Position and momentum representations: ψ(x) = hx|ψi ; ψ(p) = hp|ψi ;˜ xˆ|xi = x|xi , hx|yi = δ(x − y) , 1 =

Z

dx |xihx| , xˆ†= xˆ pˆ|pi = p|pi , hq|pi = δ(q − p) , 1 =

Z dp |pihp| , pˆ†= pˆ 1 hx|pi = √ 2π~exp ipx ~
; ψ(p) =˜ Z dxhp|xihx|ψi = √1 2π~ Z dx exp −ipx ψ ~
(x) hx|pˆn|ψi = ~_idxd nψ(x) ; _hp|ˆxn_{|ψi =}i~ d dp n ˜ ψ(p) ; [ˆp, f (ˆx)] = ~f′(xˆ) i 1 ∞ 2π Z exp(ikx)dx = δ(k) −∞

• Generalized uncertainty principle

∆A ≡ |(A − hAi1)Ψ → (∆A)2 = hA2i − hAi2 1 ∆A ∆B ≥ hΨ|_2i[A, B]|Ψi   ~  ∆x ∆p ≥ ₂ ∆ ∆x = √ 2 and ∆p = ~ √ 2∆ for ψ ∼ exp  −1₂x 2 ∆2  Z +∞ dx exp −∞ −ax 2
₌r π a d

Time independent operator Q : Q i

dth i = ~ h[H, Q]i ~ ∆H∆t ≥ 2, ∆t ≡ ∆Q  dhQi_dt  • Commutator identities

[A, BC] = [A, B]C + B[A, C] ,

eABe−A = eadA_{B = B + [A, B] +} 1

2[A, [A, B]] + 1

[A, [A, [A, B]]] + . . . , 3!

eABe−A = B + [A, B] , if [A, [A, B] ] = 0 , [ B , eA] = [ B , A ]eA, if [A, [A, B] ] = 0

eA+B = eAeBe−12[A,B] = eBeAe 1

2[A,B] , if [A, B] commutes with A and with B

• Harmonic Oscillator 1 Hˆ = pˆ2₊1 2m 2mω 2_xˆ2 _{= ~ω}  ˆ N + 1 , 2  Nˆ = aˆ†_aˆ aˆ = r mω 2~  ˆ x + iˆp mω  , aˆ† = r mω 2~  ˆ x − iˆp , mω  xˆ = r ~ 2mω(ˆa + ˆa †_{) , p = iˆ} r mω~ (aˆ†_{− aˆ) ,} 2 [xˆ, pˆ] = i~ , [aˆ, aˆ†] = 1 , [ , aNˆ _{ˆ ] = −aˆ ,} [N , aˆ ˆ†] = aˆ†. 1 |ni = √ (a†)n n! |0i Hˆ |ni = En|ni = ~ω 1 n + 2

aˆ†_{|ni =} √_{n + 1|n + 1i , aˆ|ni =} √_{n|n − 1i .} ψ0(x) = hx|0i =  mω π~ 1/4 exp₋mωx2 2~  . pˆ xH(t) = xˆ cos ωt + sin ωt mω pH(t) = pˆ cos ωt − mω xˆ sin ωt

• Coherent states and squeezed states Tx0 ≡ e −i 0 ~pˆ x , T x0|xi = |x + x0i |x˜0i ≡ Tx0|0i = e −i 0 ~pˆx |0i , |x˜0i = e− 1 4 x2 0 d2e x0 √_2da† ~ |0i , hx|x˜0i = ψ0(x − x0) , d2 = mω

|αi ≡ D(α)|0i = eαa†−α∗a|0i , D(α) ≡ expαa†_{− α}∗a, α = _√h ixˆ + i h i dpˆ

2 d √

C 2 ~ ∈ |αi = eαa†−α∗a|0i = e−21|α|2eα a†|0i , aˆ|αi = α|αi , |α, ti = e−iωt/2|e−iωtαi

 _γ |0γi = S(γ)|0i , S(γ) = exp − (a†a† 2 − aa)  , _{γ ∈ R} 1 |0γi = √ cosh γ exp  −1 tanh γ a†a† 2  |0i

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)

d

i~ Ψ ˆ d

dt| , ti = H(t)|Ψ, ti ↔ i~dtU(t, t0) = H(t)U(t, tˆ 0) i

ˆ

Time independent H: U(t, t0) = exp

h −_~H(t − tˆ 0) i =X n e−_~iE (t−tn 0) |nihn| hAi = hΨ, t|AS|Ψ, ti = hΨ, 0|AH(t)|Ψ, 0i → AH(t) = U†(t, 0)ASU(t, 0)

[AS, BS] = CS → [AH(t) , BH(t) ] = CH(t)

d

i~ AˆH(t) = [AˆH(t), HˆH(t)] , for AS time-independent

• Two state systems

H = h01+ h · σ = h01+ 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 = B0z+ B1 cos ωt x − sin ωt y

i
|Ψ(t)i = exp_~ωt ˆSz  exp i_~γBR· S t  |Ψ(0)i B_R = B0 ω 1 − z ω0
+ B1x

• Orbital angular momentum operators

a_{· b ≡ a}ibi, (a × b)i ≡ ǫijkajbk, a2 ≡ a · a .

ǫijkǫipq = δjpδkq − δjqδkp, ǫijkǫijq = 2δkq.

Lˆi = ǫijkxˆjpˆk ⇐⇒ L = r × p = −p × r

ˆ

Vector u under rotation: [ Li, uˆj] = i~ ǫijkuˆk =⇒ L× u + u × L = 2i~ u

Scalar S under rotation: [Lˆi, S ] = 0

u, v vectors under rotations _→ u_{· v is a scalar, u × v is a vector}  Lˆ , Lˆ_{i j} ₌ i~ ǫ_ijkLˆ_{k ⇐⇒ L × L = i L ,}~ _[Lˆ_{i, L}2_{] = 0 .} ∇2 ∂ 2 = ∂r2 + 2 r ∂ ∂r + 1 r2  ∂2 ∂θ2 + cot θ ∂ ∂θ + 1 sin2θ ∂2 ∂φ2  Lˆ2 _{= −}~2 ∂ 2 ∂θ2 + cot θ ∂ ∂θ + 1 sin2_θ ∂2 ∂φ2  ~ ˆ Lz = ∂ i ∂φ ; Lˆ± = ~e ±iφ  ±_∂θ∂ + i cot θ ∂ ∂φ  • Spherical Harmonics Yℓ,m(θ, φ) ≡ hθ, φ|ℓ, mi 1 Y0,0(θ, φ) = √ 4π ; Y1,±1(θ, φ) = ∓ r 3 8πsin θ exp(±iφ) ; Y1,0(θ, φ) = r 3 4πcos θ

• Algebra of angular momentum operators J (orbital or spin, or sum) [Ji, Jj] = i~ ǫijkJk ⇐⇒ J × J = i~ J ; [ J2, Ji] = 0

J2_{|jmi =}~2_{j(j + 1)|jmi ; Jz|jmi = ~m|jmi , m = −j, . . . , j .} 1 J± = Jx± iJy, (J )± †= J∓ Jx = 2(J++ J−) , Jy = 1 (J+ 2i − J−) [Jz, J±] = ±~ J±, [J , J2 ±] = 0 ; [J+, J−] = 2~ Jz J2 = J+J−+ Jz2− ~Jz = J−J++ Jz2+ J~ z J±|jmi = ~ p j(j + 1) − m(m ± 1) |j, m ± 1i • Angular momentum in the two-dimensional oscillator

1 a

ˆL= √

2(ˆax+ iˆay) , ˆaR = 1 √ (aˆx− iaˆy) , [aˆL, aˆ†L] = [aˆR, aˆ†R] = 1 2 J+= ~ ˆa†RaˆL, J− = ~ aˆ†LaˆR, Jz = 1₂~ ˆ(NR− NˆL) |j, mi : j = 1 2(NR+ NL) , m = 1_(N R 2 − NL) H = 0 ⊕12 ⊕ 1 ⊕ 3 _. 2 ⊕ . . • Radial equation  ~2 −_2m d 2 dr2 + V (r) + ~2_{ℓ(ℓ + 1) } uνℓ(r) = Eνℓuνℓ(r) (bound states) 2mr2 u (r) ∼ rνℓ ℓ+1, as r → 0 . • Hydrogen atom e2 En= − 2a0 1 n2 , ψn,ℓ,m(~x) = unℓ(r) Yℓ,m(θ, φ) r n = 1, 2, . . . , _{ℓ = 0, 1, . . . , n − 1 , m = −ℓ, . . . , ℓ} ~2 a0 = e2 , α = me2 _~c ≃ 1 , ~_c 137 ≃ 200 MeV–fm 2r u1,0(r) = _3/2 exp( a₀ −r/a0) 2r u2,0(r) = r 1 (2a0)3/2  − e 2a0  xp(−r/2a0) 1 u2,1(r) = √ 3 1 (2a0)3/2 r2 a0 exp(_−r/2a0)

• Addition of Angular Momentum J = J1 + J2

Uncoupled basis : |j j 2

2; m 2

1 1m2i CSCO : {J1, J2, J1z, J2z}

Coupled basis : _|j1j2; jmi CSCO : {J12, J22, J2, Jz}

j1⊗ j2 = (j1+ j2) ⊕ (j1+ j2− 1) ⊕ . . . ⊕ |j1− j2| |j1j2; jmi = X |j1j2; m1m2 m1+m2=m i hj1j2; m m2|j1j | 1 {z 2; jmi Clebsch−Gordan coefficien}t 1 J1 · J2 = (J1+J2−+ J1−J2+) + J1zJ2z 2 1 Combining two spin 1/2 : 1

2 ⊗ 2 = 1⊕ 0 | 1, 1 i = | ↑↑i , 1 | 1, 0 i = √ 2(| ↑↓i + | ↓↑i) , |0, 0i = 1 √ ( 2 | ↑↓i − | ↓↑i) |1, −1i = | ↓↓i .

MIT OpenCourseWare

http://ocw.mit.edu

8.05 Quantum Physics II

Fall 2013