1. No category

Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
2014/5/8
Fundamentals in
Nuclear Physics
原子核基礎
Kenichi Ishikawa (石川顕一)
http://ishiken.free.fr/english/lecture.html
[email protected]
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Elements of
• Special Relativity
• Quantum Mechanics
特殊相対性理理論論
量量⼦子⼒力力学
Material will be downloadable from:
http://ishiken.free.fr/english/lecture.html
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Special relativity
特殊相対性理理論論
1905
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Special relativity is derived from two principles.
① Special Principle of Relativity
特殊相対性原理理
Physical laws should be the same in
every inertial frame of reference. 慣性系
② Principle of Invariant Light Speed
光速度度不不変の原理理
There is at least one inertial frame of
reference where Maxwell’s equations
hold.
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Maxwell’s equations
1864年
∇·D=ρ
∇·B=0
∂B
=0
∇×E+
∂t
∇×H=J
1 ∂2E
2
−
∇
E=0
2
2
c ∂t
1
in vacuum
c= p
真空中
✏ 0 µ0
vacuum velocity of light is uniquely derived from
Maxwell’s equations
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Special relativity is derived from two principles.
① Special Principle of Relativity
特殊相対性原理理
Physical laws should be the same in
every inertial frame of reference. 慣性系
② Principle of Invariant Light Speed
光速度度不不変の原理理
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
motion of the light source.
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Special relativity is derived from two principles.
① Special Principle of Relativity
特殊相対性原理理
Physical laws should be the same in
every inertial frame of reference. 慣性系
② Principle of Invariant Light Speed
光速度度不不変の原理理
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.
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Relativity is used in nuclear physics primarily
through the expressions for the energy and
momentum of a free particle
運動量量
rest mass
energy
momentum
静⽌止質量量
E= p
mc2
1
p= p
1
=p
1
1
velocity
m
2
v 2 /c2
mv
v 2 /c2
速度度
v
= mc2 = M c2
= mv = M v
= v/c
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
rest mass
energy
静⽌止質量量
E= p
velocity
m
mc2
v 2 /c2
1
速度度
v
= mc2 = M c2
non-relativistic limit v ⌧ c usually applies for nuclei
1
E ⇡ mc + mv 2
2
2
•
Mass is a form of energy
nuclear energy
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
rest mass
energy
静⽌止質量量
E= p
velocity
m
mc2
v 2 /c2
1
速度度
v
= mc2 = M c2
non-relativistic limit v ⌧ c usually applies for nuclei
1
E ⇡ mc + mv 2
2
2
•
•
Mass is a form of energy
The faster the particle is, the larger its
observed mass is
The kinetic energy 運動エネルギー is also observed
as a part of the (observed) mass
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
rest mass
energy
静⽌止質量量
E= p
velocity
m
mc2
1
v 2 /c2
速度度
v
= mc2 = M c2
Any form of energy is
observed as mass.
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
energy E = p
1
mc2
v 2 /c2
E 2 = m2 c 4 + p 2 c 2
momentum p = p
1
v
pc
=
c
E
mv
v 2 /c2
nuclei: non-relativistic limit v ⌧ c
2
p
E ⇡ mc2 +
2m
p ⇡ mv
⌧p
neutrinos and ✓photons: mc
◆
2 2
m c
E ⇡ pc 1 +
2p2
especially for photons: m = 0
E = pc
✓
v⇡c 1
2 2
m c
2p2
◆
v=c
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
For two particles A and B
EA EB
c2 pA · pB
is independent of inertial frames of reference
(Lorentz invariant)
ローレンツ不不変量量
Especially
E2
c 2 p 2 = m2 c 4
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Decay
find
A!B+C
B
A
C
mC , pC
mA , p A
mB , pB
pB , pC
Energy conservation
EA = EB + EC
Momentum conservation
pA = pB + pC
In the rest frame of A
pB =
pA = 0
mA c 2 =
q
pC
p2 c2 + m2B c4 +
q
p2 c2 + m2C c4
not easy to solve ...
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
EC = E A
2
EC
EB
pC = pA
c2 p2C = (EA
pB
EB ) 2
m2C c4 = m2A c4 + m2B c4
c2 (pA
pB )2
2(EA EB c2 pA · pB )
Lorentz invariant
can be evaluated with a convenient
inertial frame of reference
In the rest frame of A
m2C c4 = m2A c4 + m2B c4
p2 =
"✓
m2A + m2B
2mA
2mA c2
m2C
q
m2B c4 + p2 c2
◆2
#
m2B c2
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Elements of
Quantum Mechanics
17
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Wave-particle duality to the Schrödinger equation
粒粒⼦子波動⼆二重性
plane wave ei(k·r
シュレーディンガー⽅方程式
t)
平⾯面波
p = k,
E=
h
= 1.055 10 34 J · s reduced Planck constant
=
2
Planck constant プランク定数
h = 6.626 10 34 J · s
how to extract p and E from
p (x, t) =
i
E (x, t) = i
x
t
(x, t) = ei(kx
t)
(x, t)
(x, t)
pˆ =
i
x
?
ˆ=i
, E
In quantum mechanics, a physical quantity is represented
by an operator (or matrix) in general.
t
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
p2
E=
+ V (x)
2m
kinetic energy
i
potential energy
運動エネルギー
i
pˆ =
x
ˆ=i
, E
ポテンシャルエネルギー
(x, t) =
t
t
波動関数
2
wave function
2
2m
x2
+ V (x)
(x, t)
the time-dependent Schrödinger equation (1D)
時間に依存するシュレーディンガー⽅方程式
3D
i
t
(r, t) =
2
2m
2
+ V (x)
(r, t)
| (x, t)|2 or | (r, t)|2 interpreted as probability density
to find the particle at x
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Solution with energy
(x, t) = (x)e
E=
i t
2
d2
+ V (x) (x) = E (x)
2
2m dx
the (time-independent) Schrödinger equation (1D)
E : energy eigenvalue
エネルギー固有値
波動関数
wave function
d
continuous functions
(x) and
dx
連続関数
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Free particle
d2
2mE
+ 2
2
dx
⾃自由粒粒⼦子
V (x) = 0
平⾯面波
plane wave
(x) = eikx ,
(x) = 0
k=
e
ikx
e
ikx
2mE
2
eikx
x
general solution
(x) = A eikx + B e
(x, t) = A ei(kx
t)
ikx
+ B ei(
kx
t)
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Barrier potential
①
reflection 反射 e
incident 入射
②
ikx
k
E=
< V0
2m
1
= eikx + Ae
eikx
transmission 透過
x = a (a > 0)
100 % reflection in classical mechanics
ikx
2
= BeKx + Ce
K=
2m(V0
transmission coefficient
T = |D| =
E)/
Kx
古典力学では100%反射される
3
= Deikx
2
透過係数
1
2
d
continuous
,
dx
eikx
V0
x=0
2 2
③
1+
V02
4E(V0 E)
sinh2 Ka
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
E=
2 2
k
< V0
2m
100 % reflection in classical mechanics
transmission coefficient
T = |D| =
透過係数
1
2
1+
V02
4E(V0 E)
sinh x =
2
sinh Ka
transmission is nonzero in
quantum mechanics
mV0 a2
2
e
x
2
トンネル効果
important in alpha decay
1.0
0.20
ex
tunneling effect
量子力学では透過がある
T
古典力学では100%反射される
=8
E/V0 = 0.8
0.8
0.15
0.6
0.10
0.4
0.05
0.2
0.2
0.4
0.6
0.8
1.0
E/V0
0
1
2
3
4
5
mV0
6
a
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Parity
パリティ（偶奇性）
2
if
d2
+ V (x) (x) = E (x)
2
2m dx
¯(x) = ( x) satisfies
V (x) = V ( x)
d2 ¯
¯(x) = E ¯(x)
+
V
(x)
2m dx2
2
in the absence of degeneracy
¯(x) = P (x)
(x) = P 2 (x)
P=1
P = -1
縮退がなければ
|P | = 1
P = ±1
( x) = (x)
( x) =
even or + parity 偶（＋）のパリティ
(x) odd or - parity 奇（−）のパリティ
Nuclear states can be assigned a definite parity, even or odd.
Important in the discussion of beta decay
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Angular momentum
L=r
角運動量
p
Lx =
p
Ly =
r
O
In the spherical coordinate
Lz =
✓
@
i~ y
@z
✓
@
i~ z
@x
✓
@
i~ x
@y
◆
@
z
@y
◆
@
x
@z
◆
@
y
@x
極座標では
Lx = i (sin
Ly = i ( cos
Lz =
L =
2
2
+
cos
tan
+
sin
tan
)
)
i
1
sin
sin
1
+
sin2
2
2
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
commutation relation
交換関係
[L2 , Lx ] = [L2 , Ly ] = [L2 , Lz ] = 0
[Lx , Ly ] = i Lz
[Ly , Lz ] = i Lx
[Lz , Lx ] = i Ly
Ylm ( , )
common eigenfunction of L2 and Lz
spherical harmonics
固有関数
球面調和関数
L2 Ylm ( , ) =
2
l(l + 1)Ylm ( , )
lz Ylm ( , ) = m Ylm ( , )
m=
l = 0, 1, 2, 3, · · ·
l, l + 1, · · · , l
1, l
(2l+1) eigenfunctions for given l
(2l+1) 個の固有状態
examples
Y00 =
1
4
Y10 =
3
cos
4
Y1,±1 =
3
sin e±i
8
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
how about spin?
Let’s use the commutation relation
[Lx , Ly ] = i Lz
[Ly , Lz ] = i Lx
[Lz , Lx ] = i Ly
as a definition of angular momentum (operator)
L = (Lx , Ly , Lz )
L=r
forget
p
p
r
O
For given l, there are (2l+1) eigenfunctions
行列
(eigenvectors)
(2l + 1) (2l + 1) matrix
固有ベクトル
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
l=1
l
l
1
l
Lz
1 0
0 0
0 0
2
···
0
0
1
l+1
l
L+ = Lx + iLy
L = Lx
iLy
0
0
0
···
0
0
0
2l · 1
0
0
···
0
1 · 2l
0
0
···
0
0
0
2 · (2l
0
···
0
0
0
0
(2l
0
···
0
1) · 2
0
0
3 · (2l
···
0
0
1)
0
0
0
(2l 2) · 3
···
0
···
···
2) · · ·
···
···
···
···
···
···
···
···
···
0
0
0
0
···
1 · 2l
0
0
0
···
2l · 1
0
0
0
0
0
···
0
0
0
0
0
2
0
2
0
0
0
2
0
0 0
0 0
2 0
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Let’s consider 2×2 matrices?
Lz
1
0
2
0
1
Lx
eigenvalues = ±
L
2
21
2
1
+1
2
Pauli matrices
x
=
0 1
1 0
2
0 2
0 0
L
2
0 1
1 0
Ly
2
0
i
spin
1
2
particle
L+
2
1 0
0 1
l=
2
1
2
0 0
2 0
i
0
パウリ行列
y
=
0
i
i
0
z
=
1 0
0 1
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