Modern Physics Assignment: Wave Equation & Lorentz Transformations

Nama : Moh. Luthfi Zainullah
NIM
: 191810201063
TUGAS FISIKA MODERN
Persamaan Gelombang
𝜕𝟐𝜙 𝜕𝟐𝜙 𝜕𝟐𝜙
1 𝜕𝟐𝜙
+
+
+
=0
𝜕𝑥 𝟐
𝜕𝑦 𝟐
𝜕𝑧 𝟐
𝑐 𝟐 𝜕𝑡 𝟐
Transformasi Lorent
𝑥 − 𝑣𝑡
𝑥′ =
𝟐
√1 − 𝑣𝟐
𝑐
y’ = y
z’ = z
𝑣𝑥
𝑐𝟐
𝑡 =
𝟐
√1 − 𝑣𝟐
𝑐
𝑡−
′
Turunkan terhadap x, y, z, t
𝜕𝑥′
=
𝜕𝑥
1
𝟐
√1 − 𝑣𝟐
𝑐
𝜕𝑦′
𝜕𝑧′
=
=0
𝜕𝑥
𝜕𝑥
𝑣
𝜕𝑡′
𝟐
𝑐
= −
𝟐
𝜕𝑥
√1 − 𝑣 𝟐
𝑐
Aturan Rantai
𝜕𝜙 𝜕𝜙 𝜕𝑥 ′
𝜕𝜙 𝜕𝑦 ′
𝜕𝜙 𝜕𝑧 ′ 𝜕𝜙 𝜕𝑡 ′
= ′
+
+
+
=
𝜕𝑥 𝜕𝑥 𝜕𝑥
𝜕𝑦 ′ 𝜕𝑥
𝜕𝑧 ′ 𝜕𝑥 𝜕𝑡 ′ 𝜕𝑥
=
1
𝟐
√1 − 𝑣𝟐
𝑐
1
𝟐
√1 − 𝑣𝟐
𝑐
𝜕𝜙
𝑣 𝜕𝜙
( ′ − 𝟐
)
𝜕𝑥
𝑐 𝜕𝑡 ′
𝜕𝜙
−
𝜕𝑥 ′
𝑣
𝑐𝟐
𝟐
√1 − 𝑣 𝟐
𝑐
𝜕𝜙
𝜕𝑡 ′
𝟐
2
𝜕 𝜙
=
𝜕𝑥 2
1
𝟐
√1 − 𝑣 𝟐
[
𝑐
𝜕𝜙
𝑣 𝜕𝜙
1
𝜕2𝜙
2𝑣 𝜕𝜙 𝜕𝜙 𝑣 𝟐 𝜕 2 𝜙
( ′ − 𝟐
) = [
( ′2 − 𝟐 ′ ′ + 4
)]
𝑣𝟐
𝜕𝑥
𝑐 𝜕𝑡 ′
𝑐 𝜕𝑥 𝜕𝑡
𝑐 𝜕𝑡 ′ 2
1 − 𝟐 𝜕𝑥
𝑐
]
𝜕𝜙 𝜕𝜙 𝜕𝑥 ′
𝜕𝜙 𝜕𝑦 ′
𝜕𝜙 𝜕𝑧 ′ 𝜕𝜙 𝜕𝑡 ′
𝜕𝜙
𝜕𝜙
(1) =
= ′
+
+
+ ′
=
′
′
′
𝜕𝑦 𝜕𝑥 𝜕𝑦
𝜕𝑦 𝜕𝑦
𝜕𝑧 𝜕𝑦 𝜕𝑡 𝜕𝑦
𝜕𝑦
𝜕𝑦 ′
𝜕2𝜙
𝜕2𝜙
=
𝜕𝑦 2
𝜕𝑦 ′ 2
𝜕𝜙 𝜕𝜙 𝜕𝑥 ′
𝜕𝜙 𝜕𝑦 ′
𝜕𝜙 𝜕𝑧 ′ 𝜕𝜙 𝜕𝑡 ′
𝜕𝜙
𝜕𝜙
(1) =
= ′
+
+
+ ′
=
′
′
′
𝜕𝑧 𝜕𝑥 𝜕𝑧
𝜕𝑦 𝜕𝑧
𝜕𝑧 𝜕𝑧 𝜕𝑡 𝜕𝑧
𝜕𝑧
𝜕𝑧 ′
𝜕2𝜙
𝜕2𝜙
=
𝜕𝑧 2
𝜕𝑧 ′ 2
𝜕𝜙 𝜕𝜙 𝜕𝑥 ′
𝜕𝜙 𝜕𝑦 ′
𝜕𝜙 𝜕𝑧 ′ 𝜕𝜙 𝜕𝑡 ′
= ′
+
+
+
=
𝜕𝑡 𝜕𝑥 𝜕𝑡
𝜕𝑦 ′ 𝜕𝑡
𝜕𝑧 ′ 𝜕𝑡 𝜕𝑡 ′ 𝜕𝑡
=
1
√1 −
𝑣𝟐
(
−𝑣
𝟐
√1 − 𝑣𝟐
𝑐
𝜕𝜙
+
𝜕𝑥 ′
1
𝟐
√1 − 𝑣𝟐
𝑐
𝜕𝜙
𝜕𝑡 ′
𝜕𝜙
𝜕𝜙
− (𝒗) ′ )
′
𝜕𝑡
𝜕𝑥
𝑐𝟐
𝟐
2
𝜕 𝜙
=
𝜕𝑡 2
1
𝑣𝟐
√
[ 1 − 𝑐𝟐
𝜕𝜙
𝜕𝜙
1
𝜕2𝜙
𝜕𝜙 𝜕𝜙
𝜕2𝜙
2
( ′ − (𝒗) ′ ) = [
−
2𝑣
+
𝑣
(
)]
′2
𝑣𝟐
𝜕𝑡
𝜕𝑥
𝜕𝑥 ′ 𝜕𝑡 ′
𝜕𝑥 ′ 2
1 − 𝟐 𝜕𝑡
𝑐
]
Substitusi
𝜕𝟐𝜙 𝜕𝟐𝜙 𝜕𝟐𝜙
1 𝜕𝟐𝜙
+
+
+
=0
𝜕𝑥 𝟐
𝜕𝑦 𝟐
𝜕𝑧 𝟐
𝑐 𝟐 𝜕𝑡 𝟐
1
[
1−
[
𝑣𝟐
𝑐𝟐
1
1−
𝑣𝟐
𝑐𝟐
(
𝜕2 𝜙
𝜕𝑥′
2
−
2𝑣 𝜕𝜙 𝜕𝜙 𝑣𝟐 𝜕2 𝜙
𝜕2𝜙
𝜕2𝜙
1
1
𝜕2𝜙
𝜕𝜙 𝜕𝜙
𝜕2𝜙
+ 4
)] +
+
− 𝟐[
( ′ 2 − 2𝑣
+ 𝑣 2 ′ 2 )] = 0
2
2
𝟐
′
′
𝟐
2
′
′
′
′
𝑐
𝑣
𝜕𝑥 𝜕𝑡
𝑐 𝜕𝑥 𝜕𝑡
𝑐 𝜕𝑡′
𝜕𝑥
𝜕𝑦
𝜕𝑧
1 − 𝟐 𝜕𝑡
𝑐
𝜕2𝜙
2𝑣 𝜕𝜙 𝜕𝜙 𝑣 𝟐 𝜕 2 𝜙
𝜕2𝜙
𝜕2𝜙
1
1 𝜕2𝜙
2𝑣 𝜕𝜙 𝜕𝜙
𝑣2 𝜕2𝜙
( ′2 − 𝟐 ′ ′ + 4
)]
+
+
−
[
(
−
+
)] = 0
2
2
2
2
𝟐
𝟐
𝟐
′
′
𝑐 𝜕𝑡 𝜕𝑥
𝑐 𝜕𝑡 ′
𝑣
𝑐 𝜕𝑡 ′
𝑐 𝜕𝑡 𝜕𝑥
𝑐 𝟐 𝜕𝑥 ′ 2
𝜕𝑥
𝜕𝑦 ′
𝜕𝑧 ′
1− 𝟐
𝑐
[
1
1−
𝑣𝟐
𝑐𝟐
1
𝜕2𝜙
2𝑣 𝜕𝜙 𝜕𝜙 𝑣 𝟐 𝜕 2 𝜙
1
1 𝜕2𝜙
2𝑣 𝜕𝜙 𝜕𝜙
𝑣 2 𝜕2𝜙
𝜕2𝜙
𝜕2𝜙
( ′2 − 𝟐 ′ ′ + 4
)] − [
( 𝟐 ′2 − 𝟐 ′
+ 𝟐 ′ 2 )] +
+
=0
2
2
𝟐
′
′
𝑐 𝜕𝑡 𝜕𝑥
𝑐 𝜕𝑡
𝑣
𝑐 𝜕𝑡
𝑐 𝜕𝑡 𝜕𝑥
𝑐 𝜕𝑥
𝜕𝑥
𝜕𝑦 ′
𝜕𝑧 ′ 2
1− 𝟐
𝑐
[
𝑣𝟐
1− 𝟐
𝑐
𝜕 2𝜙
𝑣 𝟐 𝜕 2𝜙
1
1 𝜕 2𝜙
𝑣 𝟐 𝜕 2𝜙
𝜕 2𝜙
𝜕 2𝜙
( ′ 2 + 4 ′ 2 )] − [
(
+ 𝟐 ′ 2 )] +
+
=0
𝑣 𝟐 𝑐 𝟐 𝜕𝑡 ′ 2
𝑐 𝜕𝑡
𝑐 𝜕𝑥
𝜕𝑥
𝜕𝑦 ′ 2 𝜕𝑧 ′ 2
1− 𝟐
𝑐
1
[
1−
[
(
𝑣𝟐
𝑐𝟐
𝜕𝑥
+
′2
𝑣𝟐
𝑐𝟐
2
2
𝜕𝑦
𝜕𝑧
2
2
𝜕𝑦
𝜕𝑧
𝑣 𝟐 𝜕2𝜙
1 𝜕 2 𝜙 𝑣𝟐 𝜕 2 𝜙
𝜕 𝜙 𝜕 𝜙
−
−
+
+
=0
)]
2
2
2
4
𝟐
𝟐
𝑐 𝜕𝑡 ′
𝑐 𝜕𝑡 ′
𝑐 𝜕𝑥 ′
′2
′2
𝑣𝟐
𝜕2𝜙
1
1−
𝜕2𝜙
𝜕2𝜙 𝑣 𝟐
1
𝜕 𝜙 𝜕 𝜙
( ′ 2 (1 − 𝟐 ) + ′ 2 ( 4 − 𝟐 ))] +
+
=0
𝑐
𝑐
𝑐
′2
′2
𝜕𝑥
𝜕𝑡
(
𝜕2 𝜙
2
𝜕𝑥 ′
−
𝜕2 𝜙 1
2
𝜕𝑡 ′
𝜕2 𝜙
2
𝜕𝑥 ′
+
𝑣𝟐
1
𝑐𝟐
𝑣𝟐
1− 𝟐
𝑐
𝜕2 𝜙
2
𝜕𝑦 ′
(1 − 𝑐 2 )) +
+
𝜕2 𝜙
2
𝜕𝑧 ′
−
Invarian
𝜕2 𝜙
2
𝜕𝑦 ′
1 𝜕2 𝜙
𝑐 𝟐 𝜕𝑡 ′ 2
+
=0
𝜕2 𝜙
𝜕𝑧 ′
2
=0