102

Quantum physics

ˆ

unperturbed Hamiltonian

ˆ

ψn = Enψn

(4.152)

H0

ˆ

Unperturbed

H0

ψn

eigenfunctions of H0

states

(ψn nondegenerate)

En

ˆ

eigenvalues of H0

n

integer ≥ 0

Perturbed

ˆ

perturbed Hamiltonian

Hˆ = Hˆ 0 + Hˆ

(4.153)

H

Hamiltonian

Hˆ

perturbation ( Hˆ 0)

Perturbed

E

= Ek +

ψk

|

Hˆ

ψk

Ek

perturbed eigenvalue ( Ek)

k

|

ˆ

2

−

eigenvaluesa

+

| ψk|H |ψn |

+ ...

(4.154)

||

Dirac bracket

n=k

Ek

En

Perturbed

ψk|Hˆ |ψn

ψk

perturbed eigenfunction

eigen-

ψk

= ψk +

ψn + ...

(4.155)

functions

b

Ek

−

En

(

ψk)

n=k

aTo second order.

bTo first order.

ˆ

unperturbed Hamiltonian

Unperturbed

H0

ˆ

ψn

ˆ

stationary

ψn

(4.156)

eigenfunctions of H0

H0ψn = En

En

ˆ

states

eigenvalues of H0

n

integer ≥ 0

ˆ

perturbed Hamiltonian

Perturbed

Hˆ (t) = Hˆ 0 + Hˆ (t)

H

(4.157)

Hˆ (t)

perturbation ( Hˆ 0)

Hamiltonian

t

time

Schrodinger¨

[Hˆ 0 + Hˆ (t)]Ψ(t) = i¯h

∂Ψ(t)

(4.158)

Ψ

wavefunction

∂t

equation

ψ0

initial state

Ψ(t = 0) = ψ0

(4.159)

¯h

(Planck constant)/(2π)

Perturbed

Ψ(t) =

cn(t)ψn exp(

−

iEnt/¯h)

(4.160)

wave-

n

cn

probability amplitudes

functiona

where

−i

t

Hˆ (t )

E0)t /¯h] dt

cn =

ψn

ψ0 exp[i(En

−

(4.161)

¯h 0

|

|

Fermi’s

2π

Γi→f

transition probability per

| ψf|Hˆ |ψi

|2ρ(Ef)

unit time from state i to

golden rule

Γi→f =

(4.162)

state f

¯h

ρ(Ef )

density of final states

4.7 High energy and nuclear physics

103

Nuclear decay

N(t) = N(0)e−λt

N(t) number of nuclei

(4.163)

remaining after time t

law

t

time

ln2

(4.164)

λ

decay constant

Half-life and

T1/2 =

λ

T1/2

half-life

mean life

T = 1/λ

(4.165)

T

mean lifetime

Successive decays 1 → 2 → 3 (species 3 stable)

4

N1(t) = N1(0)e−λ1t

(4.166)

N1

population of species 1

N1(0)λ1(e−λ1t − e−λ2t)

N2(t) = N2

(0)e−λ2t +

(4.167)

N2

population of species 2

N3(t) = N3

λ2 − λ1

1 +

−λ2

−

λ1

−

N3

population of species 3

(0) + N2(0)(1 − e−λ2t) + N1(0)

λ2

decay constant 2

→

3

λ1e

λ2t

λ2e

λ1t

λ1

decay constant 1

2

−

(4.168)

→

v

velocity of α particle

v3 = a(R − x)

Geiger’s lawa

(4.169)

x

distance from source

a

constant

R

range

Geiger–Nuttall

logλ = b+ clogR

(4.170)

b, c

constants for each

rule

series α, β, and γ

Liquid drop modela

N

number of neutrons

A

mass number (= N + Z)

aa (N − Z)2

B = avA

asA2/3

ac

Z2

+ δ(A)

B

semi-empirical binding energy

−

−

A1/3

−

A

(4.171)

Z

number of protons

av

volume term ( 15.8MeV)

+apA−3/4

Z, N both even

as

surface term ( 18.0MeV)

δ(A)

apA−3/4

Z, N both odd

(4.172)

ac

Coulomb term ( 0.72MeV)

−

aa

asymmetry term ( 23.5MeV)

0

otherwise

ap

pairing term ( 33.5MeV)

M(Z,A) atomic mass

Semi-empirical

M(Z,A) = ZMH + Nmn − B

(4.173)

MH

mass of hydrogen atom

mass formula

mn

neutron mass

σ(E) =

π

ΓabΓc

(4.174)

σ(E) cross-section for a+ b → c

g

k

incoming wavenumber

Breit–Wigner

k2

(E

E )2

+ Γ2/4

formula

a

−

0

g

spin factor

2J + 1

(4.175)

E

total energy (PE + KE)

g = (2sa + 1)(2sb + 1)

E0

resonant energy

Γ

width of resonant state R

Total width

Γ = Γab + Γc

(4.176)

Γab

partial width into a+ b

Γc

partial width into c

τ

resonance lifetime

Resonance

τ =

¯h

(4.177)

J

total angular momentum

lifetime

Γ

quantum number of R

sa,b

spins of a and b

dσ

di erential collision

dΩ

dr

cross-section

∞ sinKr

2

Born scattering

dσ

2µ

reduced mass

2

µ

formulab

dΩ

=

¯h2

0

Kr

V (r)r

K

=

kin

−

kout

|

(see footnote)

(4.178)

r

|

radial distance

V (r) potential energy of interaction

Mott scattering formulac

!

2

2

¯h

(Planck constant)/2π

dσ

α

2

χ

χ

Acos

α

lntan2 χ

=

csc

4

+ sec

4

+

¯hv

2

α/r

scattering potential energy

dΩ

4E

2

2

sin2 χ cos χ

χ

scattering angle

(4.179)

v

closing velocity

dσ

α

2 4

−

3sin2

χ

(A =

1, α

v¯h)

(4.180)

A

= 2 for spin-zero particles, = −1

for spin-half particles

dΩ 2E !

−

sin4 χ

Klein–Gordon

ψ wavefunction

equation

∂2ψ

2

2

(massive, spin

(

− m

)ψ

=

(4.181)

t

time

∂t2

zero particles)

m particle mass

Weyl equations

∂ψ

∂ψ

∂ψ

∂ψ

ψ spinor wavefunction

(massless, spin

=

σ

+ σ

+ σ

(4.182)

σ

Pauli spin matrices

1/2 particles)

∂t

±

x ∂x

y ∂y

z ∂z

i

(see page 26)

(iγ

µ

∂µ− m)ψ = 0

(4.183)

i

i2 = −1

Dirac equation

γµ Dirac matrices:

1

0

(massive, spin

where

∂µ =

∂

,

∂

,

∂

,

∂

(4.184)

γ0

= 02

−12

= 14

γ

1/2 particles)

(γ0)2 = 14 ;

(γ1)2

= (γ

2)2

= (γ3)2

(4.185)

= −σi

0

∂t

∂x

∂y

∂z

−

i

0

σi

1n n× n unit matrix