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
MIT OpenCourseWare
http://ocw.mit.edu
8.05 Quantum Physics II
Fall 2013
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