Fundamentals in Nuclear Physics 原子核基礎

Fundamentals in Nuclear Physics

原子核基礎

Kenichi Ishikawa (

)

http://ishiken.free.fr/english/lecture.html

[email protected]

Elements of

Material will be downloadable from:

• Special Relativity

• Quantum Mechanics

特殊相対性理論

Special relativity

特殊相対性理論

1905

① Special Principle of Relativity

Physical laws should be the same in

every inertial frame of reference.

② Principle of Invariant Light Speed

There is at least one inertial frame of

reference where Maxwell’s equations

hold.

Maxwell’s equations

1864

∇ · D = ρ

∇ × H = J

∇ × E + ∂ B

∂ t = 0

∇ · B = 0

in vacuum

真空中

1 c

∂

E

∂ t

− ∇

E = 0 c = 1

p ✏

µ

vacuum velocity of light is uniquely derived from

Maxwell’s equations

① Special Principle of Relativity

Physical laws should be the same in

every inertial frame of reference.

② Principle of Invariant Light Speed

There is at least one inertial frame of reference where Maxwell’s equations hold.

light in vacuum propagates with the speed c in one inertial frame of

reference, regardless of the state of

① Special Principle of Relativity

Physical laws should be the same in every inertial frame of reference.

Special relativity is derived from two principles.

慣性系

② Principle of Invariant Light Speed

There is at least one inertial frame of reference where Maxwell’s equations hold.

light in vacuum propagates with the speed c in any inertial frame of

reference, regardless of the state of

motion of the light source.

Relativity is used in nuclear physics primarily through the expressions for the energy and momentum of a free particle

運動量

rest mass

m velocity

^v

E = mc

p 1 v

/c

= mc

= M c

energy

momentum p = mv

p 1 v

/c

= mv = M v

= 1

p = v/c

rest mass

m velocity

v E = mc

p 1 v

/c

= mc

= M c

energy

non-relativistic limit usually applies for nuclei ^v ⌧ c E ⇡ mc

+ 1

2 mv

• Mass is a form of energy

nuclear energy

rest mass

m velocity

v E = mc

p 1 v

/c

= mc

= M c

energy

non-relativistic limit usually applies for nuclei ^v ⌧ c E ⇡ mc

+ 1

2 mv

• Mass is a form of energy

• The faster the particle is, the larger its observed mass is

The kinetic energy

is also observed

rest mass

m velocity

v E = mc

p 1 v

/c

= mc

= M c

energy

Any form of energy is

observed as mass.

energy E = mc

momentum p 1 v

/c

p = mv

p 1 v

/c

E

= m

c

+ p

c

v

c = pc E

nuclei: non-relativistic limit ^v ⌧ c E ⇡ mc

+ p

2m p ⇡ mv

neutrinos and photons: ^mc _⌧ ^p

v ⇡ c

✓

1 m

c

2p

◆

especially for photons: ^{m = 0}

E = pc v = c E ⇡ pc

✓

1 + m

c

2p

◆

For two particles A and B

is independent of inertial frames of reference (Lorentz invariant)

ローレンツ不変量

Especially

E

E

c

p

· p

E

c

p

= m

c

Decay A ! B + C

A

B ^C

m

, p

m

, p

m

, p

Energy conservation ^E

^{= E}

^{+ E}

Momentum conservation ^p

^{= p}

^{+ p}

In the rest frame of A

p

= 0 p

= p

m

c

=

q

p

c

+ m

c

+ q

p

c

+ m

c

find ^p

^{, p}

not easy to solve ...

Aの静⽌系で

E

= E

E

p

= p

p

E

c

p

= (E

E

)

c

(p

p

)

m

c

= m

c

+ m

c

2(E

E

c

p

· p

)

In the rest frame of A

Lorentz invariant

m

c

= m

c

+ m

c

2m

c

q

m

c

+ p

c

p

=

"✓

m

+ m

m

2m

◆

m

# c

can be evaluated with a convenient inertial frame of reference

Elements of

Quantum Mechanics

Wave-particle duality to the Schrödinger equation

粒⼦波動⼆重性 シュレーディンガー⽅程式

e

plane wave p = k, E =

= h

2 = 1.055 10

J · s reduced Planck constant

h = 6.626 10

J · s Planck constant

how to extract p and E from ?

E ˆ = i ˆ t

p = i

x , (x, t) = e

p (x, t) = i

x (x, t) E (x, t) = i

t (x, t)

平⾯波

時間に依存するシュレーディンガー⽅程式

E ˆ = i ˆ t

p = i

x , E = p

2m + V (x)

kinetic energy

運動エネルギー potential energy

ポテンシャルエネルギー

the time-dependent Schrödinger equation (1D)

3D

or interpreted as probability density to find the particle at x

i t (x, t) =

2m

2

x

+ V (x) (x, t)

i t (r, t) =

2m

+ V (x) (r, t)

| (x, t) |

| (r, t) |

wave function波動関数

Solution with energy

(x, t) = (x)e

the (time-independent) Schrödinger equation (1D)

wave function波動関数

(x) and continuous functions E : energy eigenvalue

2

2m d

dx

+ V (x) (x) = E (x)

d

Free particle

^{V (x) = 0}

d

dx

+ 2mE

2

(x) = 0 (x) = e

, e

k = 2mE

2

plane wave^平⾯波

x

e

e

general solution ^{(x) = A e}

^{+ B e}

(x, t) = A e

+ B e

Barrier potential

x = 0 x = a(a > 0) V₀

e

e

e

① ② ③

incident

reflection

transmission

E =

k

2m < V

1

= e

+ Ae

= De

K = 2m(V0 E)/ ²

2

= Be

+ Ce

= = 1

transmission coefficient ^透過係数

E = k

2m < V

T = | D |

= 1

1 +

sinh

Ka

transmission coefficient ^透過係数

transmission is nonzero in quantum mechanics

量子力学では透過がある

tunneling effect

sinhx = e^x e ^x 2

0.05 0.10 0.15

T 0.20

mV₀a²

2 = 8

important in alpha decay

0.2 0.4 0.6 0.8

1.0 E/V₀ = 0.8

Parity

2

2m d

dx

+ V (x) (x) = E (x)

if ^{V (x) = V ( x) ¯(x) = ( x)} satisfies

2

2m

d

¯

dx

+ V (x) ¯(x) = E ¯(x)

in the absence of degeneracy

¯(x) = P (x)

(x) = P

(x) P = ± 1

P = 1 ^{( x) = (x)} even or + parity

P = -1 ^{( x) = (x)} odd or - parity

Nuclear states can be assigned a definite parity, even or odd.

L

= i ~

✓ y @

@ z z @

@ y

◆

Angular momentum

O

r

p

L = r p

In the spherical coordinate

L_x = i (sin + cos

tan )

L_y = i ( cos + sin

tan ) L_z = i

L² = ² 1

sin sin + 1

sin²

2 2

L

= i ~

✓ z @

@ x x @

@ z

◆

L

= i ~

✓

x @

@ y y @

@ x

◆

[L

, L

] = [L

, L

] = [L

, L

] = 0

[L

, L

] = i L

[L

, L

] = i L

[L

, L

] = i L

common eigenfunction of L

and L

Y

( , )

spherical harmonics

球面調和関数

L

Y

( , ) =

l(l + 1)Y

( , ) l

Y

( , ) = m Y

( , )

l = 0, 1, 2, 3, · · ·

m = l, l + 1,· · · , l 1, l

Y₀₀ = 1

4 Y₁₀ = 3

4 cos examples

Y_1,±1 = 3

8 sin e^±i

固有関数

(2l+1) eigenfunctions for given l

(2l+1) 個の固有状態

how about spin?

[L

, L

] = i L

[L

, L

] = i L

[L

, L

] = i L

Let’s use the commutation relation

as a definition of angular momentum (operator)

L = (L

, L

, L

)

O r

p

L = r p

forget For given l, there are (2l+1) eigenfunctions

(eigenvectors)

固有ベクトル

(2l + 1) (2l + 1) matrix

L_z

l

l 1

l 2

· · ·

l + 1 l

L₊ = L_x +iL_y

0 1· 2l 0 0 · · · 0

0 0 2· (2l 1) 0 · · · 0

0 0 0 3·(2l 2) · · · 0

· · · · · · · · · · · · · · · · · ·

0 0 0 0 · · · 2l ·1

0 0 0 0 · · · 0

L = L_x iL_y

0 0 0 · · · 0 0

2l ·1 0 0 · · · 0 0

0 (2l 1)· 2 0 · · · 0 0

0 0 (2l 2)·3 · · · 0 0

· · · · · · · · · · · · · · · · · ·

0 0 0 · · · 1· 2l 0

1 0 0 0 0 0 0 0 1

0 2 0

0 0 2

0 0 0

0 0 0

2 0 0

0 2 0

Let’s consider 2×2 matrices?

L_z

2

1 0

0 1 L₊

2

0 2

0 0 L

2

0 0 2 0

L_x

2

0 11 0 L_y

2

0 i

i 0

eigenvalues = ± 2

L^{2 2}1 2

1

2 + 1 1 0

0 1 l = 1 2

spin particle

x = 0 1

1 0 ^y = 0 i

i 0 ^z = 1 0

0 1

Pauli matrices