Relativistic Quantum Mechanics-

7/30/2019 Relativistic Quantum Mechanics-

http://slidepdf.com/reader/full/relativistic-quantum-mechanics- 1/76





12

ψ



S l(k0)



U (1)

GR/A p (z ) GR/A

p (ω)

G





c ≈ 3 · 108 m

s

t (x,y,z )

(x0

, x1

, x2

, x3

)

x0 ≡ ct, x1 ≡ x, x2 ≡ y, x3 ≡ z.

µ, ν

xµ = (x0, x) = (x0, x1, x2, x3).



gµν =

1 0 0 00 −1 0 0

0 0 −1 0

0 0 0 −1

.

xµ xν

xµ =3

ν =0

gµν xν xµ =3

ν =0

gµν xν

xµ

xν

xµ = gµν xν

µ, ν = 0, 1, 2, 3

a, b = 1, 2, 3

aµ

bµ

aµbµ = aµbµ = a0b0 − a · b,

a · b

aµ

aµaµ =

a0

2 − |a|2



aµaµ =

< 0 aµ

= 0 aµ

> 0 aµ

s2 = aµaµ = aµgµν aν = (ct)2 − |a|2

τ

τ =s

c=

(ct)2 − |x|2

x = const.≡

0

τ

τ

S, S ′

= s

aµaµ = 0

g



x

ct

world line of (accelerated)

light cone

(world line of a photon)

x=ct

world line of a free, massive

particle: v=x/t<c

massive particle: |slope|>1 always

L

xµ

S ′ −v v xL

s2 = x′0

x′1 x2 x3

γ =1

1 − vc

2, β =

v

c

L

(Lµν ) =

γ βγ 0 0

βγ γ 0 0

0 0 1 0

0 0 0 1

(L) = ±1.



(L) = +1 L00 ≥ 1 →

(L) = −1 L00 ≥ 1 →

(L) = +1 L00 ≤ −1 →

(L) = −1 L00 ≤ −1 →

i ∂

∂tψ(x, t) =

− 2

2m∆ + V (x)

ψ(x, t)

x0 xa, a = 1, 2, 3

→

→→ →



→

ψ(xµ)

E =

p2c2 + m2c4.

i ∂

∂tψ =

− 2c2 ∇2 + m2c4ψ,

p = i

∇.

− 2 ∂ 2

∂t2ψ =

−

2c2 ∇2 + m2c4

ψ



∂ µ =

∂

∂xµ ∂

µ

=

∂

∂xµ ,

xµ = (ct, x1, x2, x3) xµ = (ct, x1, x2, x3) = (ct, −x1, −x2, −x3)

i ∂ ∂ (ct)

= i ∂ ∂x0 = i ∂ 0

i ∂ ∂xa

= −i ∂ a, a = 1, 2, 3

pµ −→ i ∂ µ = i

∂

∂ (ct)

− ∇

− 2 ∂ 2

∂ (ct)2ψ(x, t) = (−

2 ∇2 + m2c2)ψ(x, t)

⇔

∂ µ∂ µ +mc

2

ψ(xµ) = 0

2

:= ∂ µ∂

µ

= ∂

2

0 − ∇

2

mc

ψ(x, t) = exp

− i

(Et − px)

= exp

− i

pµxµ

,

E = ±

p2c2 + (mc2)2.

p



kµ

= pµ

ψ(xµ) =

d4k√

2π4 δ 4

kµkµ −

mc

2

A(kµ)e−ikµxµ.

A(kµ) k0 < 0

E = +

p2c2 + (mc2)2 > 0

ψ(+)(x, t) = e−i

(Et− p· x)

ψ(−)(x, t) = e−i

(−Et− p·x) = e−i

(E (−t)− p·x)

E < 0

∂/∂x

∼ O( p2) p

∂/∂ (ct)

ρ = ψ∗ψ



=

2mi ψ∗

∇ψ

−ψ

∇ψ∗ .

∂

∂tρ + ∇ · = 0.

ψ∗

ψ∗

∂ µ∂ µ +mc

2

ψ = 0.

ψ

∂ µ∂ µ +

mc

2

ψ∗ = 0.

∂ µ (ψ∗∂ µψ − ψ∂ µψ∗) = 0,

∂

∂t i

2mc2 ψ∗∂

∂t ψ − ψ

∂

∂t ψ∗ ρ

+ ∇ ·

2mi ψ∗ ∇ψ − ψ ∇ψ∗

= 0.

ρ

ρ

ρ(x, t)

ρ(ω, x) = ω

mc2ψ∗ψ

ρ



12

12

E = p2c2 + (mc2)2

E 2 = c2 p2 + (mc2)2 != (cα · p + βmc2)2,

α = (αx, αy, αz), β α, β

c2( p2x + p2

y + p2z) + m2c4 = c2(α2

x p2x + α2

y p2y + α2

z p2z) + β 2m2c4

+c2 px py(αxαy + αyαx) + c2 py pz(αyαz + αzαy)

+c2 pz px(αzαx + αxαz) + mc3[ px(αxβ + βαx)

+ py(αyβ + βαy) + pz(αzβ + βαz)]



12

αx, αy, αz, β

α2i = β 2 = 1

αiα j + α jαi =: [αi, α j]+ = 0

αiβ + βαi =: [αi, β ]+ = 0

i = x,y,z

i, j = x,y,z

i = x,y,z

α, β

[M µ, M ν ]+ = 2δ µν 1

M µ = β, αxαyαz.

αx, αy, αz, β

H = c α · p + βmc2

M µ λ = ±1

µ = ν

(M µ)2 = 1

M µ ⇒ λ = ±1

2

M µ

(M µ) = 0, µ = 0, 1, 2, 3



µ = ν

M µM ν = −M ν M µ

⇔ M µM µ 1

M ν = −M µM ν M µ

⇒ (M ν ) = − (M µM ν M µ)

= − (M ν M µM µ =1

)

= − (M ν )

= 0

2

αx, αy, αz, β

0 = (M µ) =d

i=1

λi =d

i=1

(±1) ⇔

2

αx, αy, αz, β d ≥ 4 d = 2

d = 4

α =

0 σ

σ 0

, β =

1 0

0 −1

σ = (σx, σy, σz)

σx =

0 1

1 0

, σy =

0 −i

i 0

, σz =

1 0

0 −1



12

α, β

i ∂

∂tψ(xµ) = (c α · p+βmc2)ψ(xµ)

p = −i ∇.

αx, αy, αz, β ψ(xµ)

• ψ(xµ)

• ψ(xµ)

2×2 = 4 = d

d = 2 · (2S + 1)



i ∂

∂tψ = (ca · p + βmc2)ψ

1c

β

(−i β∂ 0ψ + i βαi∂ iψ + mc)ψ = 0

γ 0 = β

γ i = βαi, i = 1, 2, 3.

α, β

−iγ µ∂ µ +

mc

ψ = 0

mc

γ µ∂ µ = γ µ∂

∂xµ

γ µ =

β

β α

γ ′µ = Lµ

ν γ ν

γ µuµ = γ 0u0 − γ · u =: /u

−i/∂ +

mc

ψ = 0



12

γ

γ 0 = β (γ 0)2 = 1

γ i, i = 1, 2, 3 (γ i)2 = −1

(γ i)† = (βαi)† = αiβ = −βαi = −γ i

(γ i)2 = βαiβαi = −ββαiαi = −1

2

γ

[γ µ, γ ν ]+ = 2gµν 1

a, β γ

γ 0 = 1 0

0 1

, γ i =

0 σi

−σi 0

, i = 1, 2, 3

γ µ = Aγ µA−1,

ψ

ψ =

ψ1

ψ4

, ψ† = (ψ∗1 , . . . , ψ∗

4)



ρ = ψ† · ψ =

4

i=1

ψ∗i ψi,

∂

∂t(ψ†ψ) =

∂

∂tψ†

ψ + ψ†

∂

∂tψ

.

i

∂

∂tψ

= (−i c α · ∇ + βmc2)ψ

ψ†

i ψ†

∂

∂tψ

= (−i cψ† α · ∇ + βmc2ψ†)ψ.

ψ

−i

∂

∂tψ†

ψ = i

∇ψ†

· c αψ + mc2ψ†βψ

i ∂ ∂tψ†ψ = −i ψ†(ca) · ( ∇ψ) + ( ∇ψ†) · (c α)ψ

= − ∇ · ψ†(ca)ψ

∂

∂tρ + ∇ · = 0 ∂ µ j

µ = 0,

( j

µ

) = cρ

, (∂ µ) = ∂

∂ (ct)

∂ ∂x .

ρ = ψ†ψ

= ψ†(c α)ψ = vρ

v := c α



12

jµ

j′µ = Lµν j

ν .

• −→• −→

•• −→

ψ

E 2 = p2c2 + (mc2)2

E/c =

i ∂/∂ (ct) p = −i ∇ E − p

pµ pµ =

E

c

2

− p2 = (mc)2,



ψ

ψ

γ ψ

ψ

xµ′ = Lµν x

ν x′ = L x

ψ L

ψ′(x′) = S (L)ψ(x)

= S (L)ψ(L−1(x′))

4 × 4

−iγ µ∂ µ +

mc

ψ(x) = 0 I

−iγ µ∂ ′µ +

mc

ψ′(x′) = 0 I ′

∂ µ =∂

∂xµ∂ ′µ =

∂

∂xµ′ .

γ

γ µ

γ µ = Aγ µA−1 .



12

γ µ

γ µ′ = T (L)γ µT −1(L).

γ ′

γ ′ = γ µ ⇒ A = T −1(L)

2

∂ µ = ∂ ∂xµ

= ∂xν ′

∂xµ∂

∂xν ′ = Lν µ∂ ′ν

xν ′ = Lν µxµ

∂xν ′

∂xµ= Lν

µ

ψ ψ′

S −1ψ′(x′) = ψ(x).

−iγ µLν

µ∂ ′ν +mc

S −1ψ′(x′) = 0.

S

−iSLν µγ µS −1∂ ′ν ψ

′(x′) +mc

ψ′(x′) = 0.

S (L)

SLν µγ µS −1 = γ ν

S −1(L)γ ν S (L) = Lν µγ µ



S (L)

Lν µ = δ ν

µ + ∆ων µ,

∆ων µ

L00 = 1

Lab = cos(φ)

Lν ν = cosh(φ)

⇒ Lν ν = 1 + O(φ2)

Lν

µ ∼ sin(φ)

sinh(φ) = O(φ), ν = µ

⇒ ∆ων ν = O + O(φ2)

(∆ων µ) φ

S (L) ∆ων µ

S = 1 + τ S −1 = 1− τ τ

(1− τ )γ µ(1 + τ ) = γ µ + γ µτ − τ γ µ + O(τ 2)

= γ µ + ∆ωµν γ ν

[γ µ, τ ] = ∆ωµν γ ν



12

τ 1

τ 1, τ 2

[γ µ, τ 1 − τ 2] = 0.

τ 1 − τ 2 = α · 1 α ∈ R

2

τ

• S (L)

ψ†(x)ψ(x) =4

α=1

ψ∗α(x)ψα(x)

ρ =

ψ†(x)ψ(x)x ρ

(S ) = 1,

1 = (S ) = (1 + τ ) = (1) + (τ )

= 1 + (τ ) + O(τ 2)

(τ ) = 0

τ =1

8∆ωµ

ν ′gν ′ν (γ µγ ν − γ ν γ µ) =

1

8∆ωµ

ν ′gν ′ν [γ µ, γ ν ]



Lν µ = δ ν

µ + ∆ων µ

S (L) = 1 +1

8∆ωµ


•

(ων µ)

δ ν

µ +

η

N ων

µ, N → ∞

η

Lν µ =

lim

N →∞

1 +

η

N ωN ν

µ

= (eηω )ν µ

−β = −v/c

ω =

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

=: (τ 01)x 101 :=

1 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0



12

(Lν µ) = 1 +

∞k=1

1

k! (ητ x1)k

= 1 − 101 +∞

k=0

1

(2k)!η2k101

+∞

k=0

1

(2k + 1)!η2ki+1(τ 01)x

(Lν µ) = 1 − 101 + cosh(η)101 + sinh(η)(τ 01)x

= cosh(η) sinh(η) 0 0

sinh(η) cosh(η) 0 00 0 1 0

0 0 0 1

η

v/c

tanh(η) =v

c= β,

cosh(η) = γ =1

1 − vc2

,

sinh(η) = βγ.

•

S (L) = limN →∞

[1 +η

N

1

8(ωµ


(∗)

)]N

= expη · 18

(ωµν ′g

ν ′ν [γ µ, γ ν ])∗ 4×4



−v/c x

ωµν ′g

ν ′ν [γ µ, γ ν ] =

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

µν

× [γ µ, γ ν ]

= 4α

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

=

0 −1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

[γ µ, γ ν ] = [γ µ, γ ν ]+− 2γ ν γ µ

= 2gµν 1

−2γ

ν γ

µ

[γ 0, γ 1] = −2γ 1γ 0

= −2

0 −σ1

σ1 0

1 0

0 −1

= −2

0 σ1

σ1 0

= −2α1

S (L)

S (L) = expη

2α1

=∞

k=0

1

(2k)!

η

2

2k

+∞

k=0

1

(2k + 1)!

η

2

2k+1

α1

S (L) = cosh(η/2)1+ sinh(η/2)α1

S (L) =

cosh(η/2) 0 0 sinh(η/2)

0 cosh(η/2) sinh(η/2) 0

sinh(η/2) 0 0 cosh(η/2)



12

x tanh(η) = v/c

•

jµ = c ψ†γ 0 ψ

γ µψ = cψ†

1

α

ψ

S †

S −1

S †γ 0

= bγ 0

S −1

b =

+1, L00 ≥ 1

−1, L00 ≤ −1

ψ := ψ†γ 0,

jµ

ψ′ = Sψ

ψ′ = ψ†S †γ 0 = bψ†γ 0S −1 = bψS −1

jµ = cψγ µψ

jµ′ = cbψ S −1γµS

=Lµ

νγ ν

ψ = cbLµν ψγ ν ψ = bLµ

ν jν



ψψ = ψ†γ 0ψ

ψ′ψ′ = bψS −1Sψ = bψψ

i

∂

∂t ψ = [c α · p + βmc

2

]ψ

ψ(x) −→ ψ′(x) = e−iθ(x)

ψ(x)∂

∂ (ct)−→ ∂

∂ (ct)+ i

∂θ

∂ (ct)=: Dt

∂

∂x−→ ∂

∂x+ i

∂θ

∂x=: Dx

pµ = i ∂

∂xµ−→ pµ −

∂θ

∂xµ= Πµ

x = ctx .

(∂θ/∂xµ)

Πµ = pµ − q

cAµ



12

q

cAµ =

∂θ

∂xµ

q

Πµ = gµν Πν = pµ − q

cAµ

cΠ0 = i ∂

∂t− q Φ

Π = −i

∂

∂x −q

c A

i ∂

∂tψ = [c α ·

p − q

c A

Π

+βmc2 + q Φ]ψ

v ≪ c Φ = 0

Eψ = c α · Π + βmc2ψ.

ψ(t) = ψe−i

Et

ψ(x) =

χ(x)

Φ(x)

,

χ, ΦE − mc2 −cσ · Π

−cσ · Π E + mc2

χ

Φ

= 0

(E − mc2) E s

χ − cσ · Π = 0

(E + mc2)

E s+2mc2

Φ − cσ · Πχ = 0



E s mc2

E

v ≪ c q

c A ≪ | p|

E s ≈ p2

2m≪ mc2

| Π| ≈ m|v| ≪ mc

Φ =

cσ · Π

E + mc2

χ

E + mc2 = E s + 2mc2

≈2mc2

c · | Π| ≈ mc|v|Φi

χi

∼= 1

2

v

c≪ 1, i = 1, 2

Φ ∼= 1

2mc(σ · Π)χ

χ Φ

E sχ =1

2m(σ · P i)(σ · P i)χ

E s

−→i

∂

∂t

i ∂

∂tψ =

1

2m(σ · Π)(σ · Π)ψ

(σ · u)(σ · v) = u · v + iσ · [u × v]



12

u v

[ Π × Π] = p −q

c A× p −q

c A= [ p × p]

=0

−q

c[ A × p] − q

c[ p × A] +

q 2

c2[ A × A]

=0

= −q

c[ A × p] − q

c[ p × A] +

q

c[ A × p]

=i q

c[ ∇ × A]

=i q

c B

i ∂

∂tχ =

1

2m( p − q

c A)2 − q

mc =2µB

·1

2σ · B

χ

12

σqmc

= 2µB = gµB

1

2 g = 2

µB =q

mc

= c = 1

(−

iγ µ∂ µ + m)ψ = 0

m

, c

m −→ mc

=

1

λ

−→ mc2 = E 0



pψ = 0

(−iγ 0∂ 0 + m)ψ = 0

γ

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

(−i∂ 0) +

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

m

ψ = 0

ψ(+)(x) = ur(E = m, p = 0)e−imt r = 1, 2

ψ(−)(x) = vr(E = m, p = 0)e+imt m =

u1(m, 0) =

1

0

0

0

, u2(m, 0) =

0

1

0

0

v1(m, 0) =

0

0

1

0

, v2(m, 0) =

0

0

0

1

( pµ) =

m 0

E > 0(u1, u2) E < 0(v1, v2)

p

−v

x p px = p

p v



12

E p = γ βγ

βγ γ m

0 = γm

βγm

γ =1

1 − β 2

β =v

c= v

px = βγ m

γm

L µ

x ν

=

cosh η sinh(η) 0 0

sinh(η) cosh(η) 0 0

0 0 1 0

0 0 0 1

.

S (Lx

) = cosh(η/2)1 + sinh(η/2)α1,

u′1(E, p) =

cosh(η/2)

0

0

sinh(η/2)

, u′2(E, p) =

0

cosh(η/2)

sinh(η/2)

0

v′1(E, p) = 0

sinh(η/2)cosh(η/2)

0

, v′2(E, p) = sinh(η/2)

00

cosh(η/2)

u′i, v′i, i = 1, 2 E

p

cosh(η) = γ =E

msinh(η) = βγ =

px

m(E > 0)



1/2

cosh(η/2) = 12

(cosh(η) + 1) = 12m

(E + m)

sinh(η/2) = (η)

1

2(cosh(η) − 1) = ( px)

1

2m(E m)

= ( px)

1

2m

E 2 − m2

E + m= px

1

2m(E + m)

u1(E, px) =

1

2m(E + m)χ1

px

1

2m(E +m)χ2

u2(E, px) =

12m

(E + m)χ2

px

1

2m(E +m)χ1

v1(E, px) =

px

1

2m(E +m)χ2

12m

(E + m)χ1

v2(E, px) =

px

1

2m(E +m)χ1

1

2m(E + m)χ2

χ1 =

1

0

χ2 =

0

1

.

v ≪ c

u1, u2 ≈ 1,

u1, u2

≈v

2c ≪1,

v1, v2 ≈ v

2c≪ 1,

v1, v2 ≈ 1.

p

pxχ2 −→ σ · pχ1

pxχ1 −→ σ · pχ2.



12

px

e−imt′ = e−ip0′x0′ = e−ipµxµ

= e−i(Et− px)

e+imt′ = e+ip0

′x0′ = e+ipµxµ

= e−i(−Et+ px)

p

ψ(+) p,r (x) = ur(E, p)e−i(Et− px)

ψ(−) p,r (x) = vr(E, p)e−i(−Et+ px)

E = +

p2 + m2

H 2D = −i ∂ ∂t

H D

ur, vr

ur(k)us(k) = δ rs r, s = 1, 2

vr(k)vs(k) = −δ rs

ur(k)vs(k) = 0

vr(k)us(k) = 0

Lµν = δ µν + ∆ωµ

ν

S (L) = 1 +1

8∆ωµν [γ µ, γ ν ]



∆ ϕ, |∆ ϕ| =

∆ωij = −εijk ∆ϕk (∆ω0µ = ∆ωµ

0 = 0)

σµν =i

2[γ µ, γ ν ]

σij = σij = εijk Σk

Σk = σk 0

0 σ

k σk

S (L) = 1− i

4∆ωµν σµν

ψ′(x′) = L {ψ(x)} = Sψ(x) = Sψ

L−1x′

β ±1

E < 0

E > 0

E < 0



12

ψ(x) =

d4 p

(2π)4δ ( p2

0 − E 2)

s=1,2

(2π)2mb( p, s)W s( p) e−ipµxµ

d4 p(2π)4

δ ( p20 − E 2) = d3 p

E

ψ

2m

b( p, s) E =

| p|2 + m2

W s( p) =

us( p), p0 > 0

vs( p), p0 < 0.

δ ( p20 − E 2) = 1

2 p0[δ ( p0 − E ) + δ ( p0 + E )]

ψ(x) = d3 p(2π)3

mE

s=1,2

b( p, s)us( p)e−ipµxµ + d∗( p, s)vs( p)e+ipµxµ

b( p, s) = 2πb(E, p, s)

d∗( p, s) = 2π

b(−E, p, s).

E > 0

E > 0

d∗ ≡ 0 p0 = E

ψ(+)(x) =

d3 p

(2π)32

m

E

s=1,2

b( p, s)us( p)e−ipµxµ



J (+) =

d3x

(2π)3 (+)(x)

= c

d3x

(2π)32

ψ(+)†(x) αψ(+)(x)

= c

s

d3 p

(2π)3

m

E

b( p, s)2 p

E

= p

E =

vG

vG = ∂E ∂ p

=∂ √ | p|2+m2

∂ p= p

E .

u v

E > 0 E < 0

E > 0 E < 0

t = 0

E > 0 d

4d

ψ(t = 0, x) =1

(2πd2)34

eix k− x2

(2d)2 w

w E > 0 w =

χ1

0

=

1

0

0

0



12

4d

k

eix k− x2

(2d2) =

d3 p

(2π)3

4πd2

2π

32

e−d2( p− k)2

p k

b( p, s) = 232 d3e−d2( p− k)2u†s(vp)w = 0

d∗( p, s) = 232 d3e−d2( p− k)

2

v†s( p)w = 0

ψ(t, x) E > 0 E < 0

E < 0 E > 0

χC

d ≪ χC =

mc

,c=1=

1

m.

w =

χ1

0

us vs

d∗( p)

b( p)=

| p − k|E + m

| k| ≪ 1

.



• d ≫ 1m

| p − k| d−1 ≪ m ⇐⇒ d∗

b≪ 1.

• d ≪ 1m

| p − k| ≈ d−1 ≫ m =⇒ | p − k| ≈ E ⇐⇒ d∗

b≈ 1

x

E > 0

x =

d3x ψ†(x)xψ(x)

d

dtx =

d

dt

d3x ψ†(0, x)e+iHtxe−iHt ψ(0, x)

=

d3x ψ†(t, x)i [H, x]

−ic α

ψ(t, x)

=

d3x ψ†(x)c αψ(x) = J (t)

J i(t) = d3 p

(2π)3

m

E pi

E s |b( p, s)

|2 +

|d( p, s)

|2

+ is,s′

b∗( p, s)d∗( p, s)e2iEtus( p)σi0vs′( p)

− b( p, s)d( p, s)e−2iEtvs′( p)σi0us( p)

2E > 2mc2

= 2 × 1021s−1



12

I II

x1=x

V(x1)=q (x1)φ

V0

V (x1) = const

E

E +m

2m

E > 0

ψin(x) = e−iEteipx

1

0

0 p

E +m

ψrefl(x) = e−iEtae−ipx

1

0

0− pE +m

u1

+be−ipx

0

1

− p

E +m

0

u2

E = +

p2 + m2 > m, p = +√

E 2 − m2.



ψrefl

ψtrans(x) = eiEt

ceiqx

1

0

0q

E −V 0+m

+ deiqx

0

1q

E −V 0+m

0

q =

+

(E − V 0)2 − m2, |E − V 0| ≥ m

i

m2 − (E − V 0)2, |E − V 0| ≤ m.

E − V 0 + m

q = 0

E

q

Re(q)Re(q)

Im(q)

m+V0m−V0 V0

•

E ≥ m + V 0 > 0 E ≤ −m + V 0 ( E ≥ m : V 0 ≤ 2m)



12

•

−m + V

0< E < m + V

0

∂ ∂xµ

ψI (0)!

= ψII (0)

ψ

1 + a = c

b = d

−b pE +m

= d qE −V 0+m

2.⇐⇒ b = d = 0

V 0 = 0 , p = q

(1 − a) = rc , r = q p

E +mE −V 0+m



propagating

solutions

m0

E>m

E

x

E−V0<−mE<−m

(II)(I)

(2)

−m

E−V0>m

V0+mV +m

V0−m

m

(1)

0

m

• |E | > m

• |E

−V 0

|> m

E > m

V 0 > 0

1) E − V 0 ≥ m V 0 > 0 E − V 0 > 0

2)

E

−V 0

≤ −m

E ≥ m m

≤E

≤V 0

−m

E 0 > 2m E − V 0 < 0.

E ≥ m V 0 ≥ 2m

•V 0

•



12

jtrans ⊥ jrefl

ψ x = 0

c =2

1 + ra =

1 − r

1 + r.

j = cψ† αψ α = 0 σ

σ 0 c

j ψ

jy = jz, jx = 0.

jtrans

jin

=4r

(1 + r)2

,jrefl

jin

= 1 − r

1 + r2

jtrans

jin

+jrefl

jin

= 1.

q,p > 0, m < E < V 0 − m, i.e. V0 > 2m, E − V0 + m < 0, r < 0

jrefl

jin

> 1,jtrans

jin

< 0

vtrans,x =dE

dq =

d

dq

q 2 + m2

=

2q

+

q 2 + m2=

2q

E − V 0= − 2q

|E − V 0|

vtrans



E > m V 0 > 2m

jtrans

E < 0

E < 0

• E < 0

• E < 0

E < 0E > 0

E < 0

e−

e+

hω

p

EE(p)



12

E < 0

E < 0, p, q

E > 0, − p, − q ; − σ

ω > 2mc

2

hω

p



E < 0

-4 -2 0 2 4-3

-2

-1

0

1

x

E < 0

ψ(±) ≡ 0

• ψ(+)(x) E > 0 p σ E > 0 p σ e

• ψ(−)(x) E < 0 p σ −E > 0 − p −σ −e

particle

particle

antiparticle

0

V

x

V

jin(+)

(+)

jrefl>jin

ψ (−)

jtrans<0

V0>2m

m<E<V0−0

ψ

ψ



12

•

••

∆x

∆ p ≥ ∆x−1

∆E ≈ c ∆ p ≥ c

∆x

=⇒ ∆x <1

2

mc=

1

2χC

∆E ≥ 2mc2

χC

E < 0

E < 0 e = −e0 < 0 E > 0

−e = e0 > 0



[γ µ(i∂ µ − eAµ) − m] ψ = 0

[γ µ(i∂ µ + eAµ) − m] ψC = 0

ψ ψC ψC = C ψ

i∂ µ eAµ

C [−γ

µ

∗(i∂ µ + eAµ) − m] C−1C ψ∗ = 0.

γ µ −→ γ µ∗ CC γ µ∗ C−1 = −γ µ

γ µ

γ µ

γ 0 = β =

1 0

0 −1

, γ i =

0 σi

−σi 0

, i = 1, 2, 3 ,

C•

•σi

−→σi∗ (i = 1, 2, 3)

C =

0 0 0 1

0 0 −1 0

0 −1 0 0

1 0 0 0

= iγ 2

C−1 =

C .



12

C ψ = C ψ∗ = iγ 2 ψ∗

γ µγ ν + γ ν γ µ = 2gµν 1

(iγ

2

)γ

µ

∗(iγ

2

) = [(−i

−γ 2

γ

2

∗ ) =iγ 2

γ

µ

(−iγ

2

∗)]∗

= −[γ 2γ µγ 2]∗

= −[γ 2(2gµ21− γ 2γ µ)]∗

= [2δ µ2γ 2 − γ µ]∗

=

−γ µ , µ = 2

2γ 2∗ − γ 2∗ = −γ 2 , µ = 2

(γ 2)2 = −1

• ψ C• E −→ −E ψ ( )∗

• p −→ − p

• (C)

• e −e



ψ =

1

0 p√

E +m

0

e−i(Et− pz)

C ψ =

0 0 0 1

0 0 −1 0

0

−1 0 0

1 0 0 0

1

0 p√

E +m

0

e+i(Et− pz)

=

0− p√ −E −m

0

1

e−i(−Et−(− p)z)

E > 0 p|| z pz = p

↑E < 0 p || z pz = − p ↓

Aµ

T : t −→ −t



12

T : p −→ − p ( )

L = [r × p] −→ − L ( )

S −→ − S ( L)

Φ(x, t) −→ Φ(x, −t) = Φ(x, t)

A(x, t) −→ A(x, −t) = − A(x, t)

e −→ e ( )

(1.264),(1.265)=⇒ Aµ(x, t) −→ Aµ(x, t) ( )

[γ µ(i∂ µ − eAµ) − m] ψ = 0

[γ µ(−i∂ µ − egµν Aν ) − m] ψT = 0

ψT = T ψ

ψT = T ψ = iΣ2ψ∗

iΣ2 =

0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

= i

σ2 0

0 σ2

.

C ≡ T



P : x −→ −x t −→ t , E −→ E

p −→ − p

L −→ + L

S −→ + S

Aµ −→ Aµ

Φ −→ Φ A −→ − A

e−ipµxµ

P : p0 −→ p0 , x0 −→ x0

pi −→ − pi , xi −→ −xi , i = 1, 2, 3.

P ψ(x, t) = γ 0 ψ(−x, t) γ 0 =

1 0

0 −1

.

Σ =

2

σ 0

0 σ



12

ψ(+) p=0 , ↑ =

1

0

0

0

e−imt , ψ(+) p=0 , ↓ =

0

1

0

0

e−imt , E = m > 0

ψ(−) p=0 , ↑ =

0

0

1

0

e+imt , ψ(−) p=0 , ↓ =

0

0

0

1

e+imt , E = −m < 0

Σz =

2

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1



•

z

ψ(+) p|| b z , ↑(x) =

E + m

2m

1

0 pz

E +m

0

e−ipµxµ

ψ(+)

p|| b z , ↓

(x) = E + m

2m

0

1

0 pz

E +m

e−ipµxµ

ψ(−) p|| b z , ↑(x) =

E + m

2m

pz

E +m

0

1

0

e+ipµxµ

ψ(−) p|| b z , ↓(x) = E + m

2m 0

pz

E +m0

1

e+ipµxµ

z||p^ ^

Σz

Σz s = ±12



12

• ⊥

z

ψ(+) p|| b x , ↑(x) =

E + m

2m

1

0

0 pz

E +m

e−ipµxµ

ψ(+) p|| b x , ↓(x) =

E + m

2m

0

1 pz

E +m

0

e−ipµxµ

ψ(−) p|| b x , ↑(x) =

E + m

2m

0

pzE +m

1

0

e+ipµxµ

ψ(−) p|| b x , ↓(x) =

E + m

2m

pzE +m

0

0

1

e+ipµxµ

z

x||p^ ^

^

1 i∂

∂tψ =

m + eΦ α · π

ւ

ր α · π −m + eΦ

ψ = H ψ.

γ µ

γ µ = A γ µ A−1



E < 0

E < 0

ψ

••

Σz σz

σz = ±m

E

2!

|v| → c E =

p2 + m2 → ∞

σz px→∞ = 0 σy px→∞ = 0.

|v| → c

x

v=c

γ µ

ψ(+)

↑ ↓ ↑ ↓



12

ψ(+)† p|| b x , ↑ · ψ(+)

p|| b x , ↑ =E + m

2m 1 0 0 pxE +m

1

0

0 pxE +m

=E + m

2m

(E 2 + m2 + 2Em) +

E 2−m2 p2

(E + m)2

=E + m

2m

2E 2 + 2Em

(E + m)2=

E

m

d4x

σz =

normalization m

E ψ

(+)† p|| b x , ↑ Σz ψ

(+) p|| b x , ↑

=m

E

E + m

2m

(E 2 + m2 + 2Em) − p2

(E + m)2

=m

E

E + m

2m

2m2 + 2Em

(E + m)2=

m

E .



ψ(+) p , ↑(x) =

E + m

2m

1

0 pz

E +m px+ipy

E +m

e−ipµxµ

ψ(+) p , ↓(x) = E + m2m 0

1 px+ipyE +m

pzE +m

e−ipµxµ

ψ(−) p , ↑(x) =

E + m

2m

pz

E +m px+ipy

E +m

1

0

e+ipµxµ

ψ(−) p , ↓(x) =

E + m

2m

px+ipy

E +m pzE +m

0

1

e+ipµxµ

E −mm

A−1 =: e−iS e−iS



12

ψ′ = e+iS ψ

i∂

∂te−iSe+iS

1

ψ = H e−iSe+iS 1

ψ

i∂

∂t

e−iS ψ′

i( ∂ ∂t

e−iS)ψ′+ie−iS ∂ ∂t

ψ′

= H e−iS ψ′ e+iS ·

i∂

∂tψ′ = e+iS

H − i

∂

∂t

e−iS

H′

ψ′.

H′

H = α · p + βm =

1m α · p

α · p −1m

= βm + O

odd

B =

Bx

0

Bz

H = σxBx + σz Bz

Θ

σ

z

x

y

ϑ0

→ ei2

σyϑ0 = e−12

σzσxϑ0



e±iS = e±

σz

β

σx

( α · p ) ϑ( p) = 1 cos ϑ ± β ( α · p )sin ϑ

= e±β O ϑ| p | ( )

p = p

| p |

H′ = eiS

H e−iS

= eβ ( α· b p )ϑ( α · p + βm)(1 cos ϑ − β ( α · p )sin ϑ)

= eβ ( α· b p )ϑ (1 cos ϑ + β ( α · p )sin ϑ) eβ( α·bp )ϑ

( α · p + βm)

= (cos 2ϑ + β ( α · p )sin2ϑ)( α · p + βm)

= α · p

cos2ϑ − m

| p |

!

= 0

+βm

cos2ϑ +

| p |m

sin2ϑ

=⇒ tan2ϑ = | p|m

sin2ϑ =p

m2 + p2=

p

E , cos2ϑ =

m m2 + p2

=m

E

[α , β ]+ = 0

H′

H′ = βm m

E + | p

|2

mE = β

1

E | p |2

+ m2

E p

| p | ≪ m

iS = β O ϑ

| p | ≈ β O 1

2m



12

e±β ( α· b p )ϑ =∞

k=0

1

(2k)![β ( α ·

p )]2k +

∞k=0

1

(2k + 1)![β ( α ·

p )]2k+1

( α · p )2 = αi piα j p j = 12{αi, α j}

δij

pi p j = | p |2

H = α ·

p − e A

+ βm + eφ = βm + ε + O

ε = 1eφ βε = ε

O = α( p − e A) β O = −Oβ

βm O(m)1m

iS =β

2mO O = α·

p − e A

.

| p−e A|m

H′ = eiSH − i ∂ ∂t e−iS

∂ ∂t

e−iS

eABe−A = B+[A , B]+1

2[A , [A , B]]+. . .+

1

k![A , [A , . . . , [A , B] . . .]]

+ . . .



H′ = H + i[S , H] +i2

2 [S , [S , H]]

+i3

6[S , [S , [S , H]]] +

i4

24

O( 1m3 )

[S , [S , [S , [S , H]]]]

− S − i

2[S , S] − i2

6[S , [S , S]] + . . .

β α 1m

i[S , H] = −O + β 2m

[O , ε] + 1m

β O2

O O 1m0

S = −i β 2m

O ≡ 0

H′ = βm +

ε′

β O2

2m − O4

8m2+ ε

− 1

8m2[O , [O , ε]] +

β

2m[O , ε] − O3

3m2 O′

= βm + ε′ + O′

O′ ∼ O 1m

O 1

miS =

β

2mO′ ∼ O

1

m2

.

H′′ = βm + ε′ +β

2m[O′ , ε′] +

O

1

m4

= βm + ε′ + O′′.



12

iS = β 2m

O′′ ∼ O 1m3

O

1m3

H′′′ = β

m +

O2

2m− O4

8m3

+ ε − 1

8m2

O , [O , ε] + iO

+ O

1

m4

O −→ H′′′ O 1

m3

O H′′′

H′′′ = β m + p

−e A

2

2m −1

8m3 p − e A2

− e Σ · B2 + eφ

− e

2mβ Σ · B − e

8m2 Σ ·

∇ × E

− e

4m2 Σ ·

E ×

p − e A

− e

8m2 ∇ · E .

H′′′

E > 0 ψ′ = χ0

i∂ϕ

∂t=

m + eφ +

1

2m

p − e A

2

− e

2mσ · B

−(| p |2)2

8m3− e

4m2σ ·

E ×

p − e A

− e

8m2 ∇ · E

ϕ

H1 = − (| p |2)2

8m3

| p |2

2

m2 + p2

=| p |2

2m− (| p |2)2

8m3+ . . .



E = − ∇φ(r) = −1

r

∂φ

∂rr , A = 0

σ ·−1

r

∂φ

∂rr × p

= −1

r

∂φ

∂r

σ · L

H2 =

e

4m2

1

r

∂φ

∂r

σ · L

∂φ∂r

H3 = − e

8m2 ∇ · E

H3

∼ λCompton =

mc

m = 0

γ µ pµ ψ = 0

1 p0 ψ = c α

· p ψ

m = 0

{αi, α j} = 2i εijk αk



E =

p2c2 + (mc2)2 m = 0

E > 0

v p =∂E ∂ p

m=0=

∂ ∂ p

(| p |c) = c p

| p | ,

| p |E

= 1

p σ⊥ = 0 m = 0

σ2⊥ > 0 σz

σx σy

m = 0 Σ p

h( p ) = Σ · p

| p |

h( p )

Σ · p

| p |2

2

=3

i,j=1

ΣiΣ j pi p j

| p |2

=

3i=1

Σ2i

=1

p2i

| p |2 = 1,

pi p j =

0 , i = j

p2i , i = j



h( p ) h = ±1

Σ

Σ

p

p helicity h=+1

helicity h=−1

Σ · p

Σ · p , (c α · p + mc2) H

= 0

Σ · p

Σ · p

γ µ pµ ψ = 0

γ · p ψ = γ 0 p0 ψ γ = 0 σ

−σ 0 −→ Σ · p Σ =

σ 0

0 σ

γ Σ

0 −11 0

0 −11 0

= γ 5γ 0 , γ 5 ≡ iγ 0γ 1γ 2γ 3 = 0 1

1 0

m = 0 γ 5γ 0

Σ · p ψ = γ 5 p0 ψ



Σ · p , γ 5

= 0

γ 5 = iγ 0γ 1γ 2γ 3

ψch = Uψ U =1√

21 + γ 5

γ µ ch = Uψ U −1 ,

p0 − σ · p

ψch

1 = 0 p0 + σ · p

ψch

2 = 0

σ · p| p |

ψch

=1

√ 21 + γ 5

σ · pP−→ −σ · p

h = +1 h = −1

h = −1

h = +1





γ

• γ 0

• γ k

γ 0 = 0 σ2

σ2 0 ,

γ 1 = i

0 σ1

σ1 0

,

γ 2 = i1 0

0−1

σ3 = i

0 σ3

σ3 0

iγ µ∂ µ − m

ψ = 0 .

ψC = ψ∗ ,

γ µ(i∂ µ − eAµ) − m

ψ = 0 | ∗ − γ µ(−i∂ µ − eAµ) − m

ψ∗ = 0

γ µ(i∂ µ + eAµ) − m

ψ∗ = 0 ψ∗ = ψC

ψ ψ∗ = ψ



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Relativistic Quantum Mechanics-