phys 342 - lecture 8 notes - f12
TRANSCRIPT
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8/11/2019 PHYS 342 - Lecture 8 Notes - F12
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Lecture 8
Summary: Mass, Momentum, Energy
Mass:
Energy:
Momentum:
!
p = m!
v = !vm
0
!
v
E = mc2= !
vm
0c2
m = !vm
0, !
v =
1
1" v2/ c
2
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Lecture 8
Energy, Momentum, and Velocity
Energy:
220
22
42022
2220
22
4202
)()(
11
cmpcE
cmcv
cvm
cv
cmE
+=
+
!
=
!
=
Velocity:
E
cp
c
v
c
v
c
E
c
vcmvmp vv
!!
!!
!!
=
$%&
=!"#$
%&
==2
2
00 ''
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Lecture 8
Total Energy, Kinetic Energy, Rest Energy
TTE
ETTEE
ETE
EEpc
)2(
2
)(
)(
0
20
20
20
20
20
20
22
+=
!++=
!+=
!=
TTEpc )2( 0 +=
E: total energy
E0: rest energyT: kinetic energy
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Lecture 8
Non-relativistic Regime
Non-relativistic regime: v
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Lecture 8
Extreme Relativistic Regime
Extreme relativistic regime: v ~ c, where Newtonianmechanics fails completely.
ET
c
Ep
mm
!
!
>> 0In other words, the rest energy is
negligible, compared to the kineticenergy.
Objects that travel at the speed of light must have zero rest
mass or rest energy and thusE = pc.
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Lecture 8
Rest Energy
In a many-body system, the rest energy of the system
includes all forms of energy except for the kinetic energy
of the system. In other words, it is not a simple summation
of m0c2.
Example: A system of two moving particles.
Tcm +2
0 Tcm +2
0
2
0
2
0
2
0
2
0
2
22
cmc
Tcmc
>
+=
The rest energy of the system is
given by:
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Lecture 8
Lorentz Transformation ofp andE
zuz
yuy
xux
u
ump
ump
ump
cmE
0
0
0
2
0
!
!
!
!
=
=
=
=
In S: In S:
zuz
yuy
xux
u
ump
ump
ump
cmE
!=!
!=!
!=!
=!
!
!
!
!
0
0
0
2
0
"
"
"
"
where
2222
1
1,
1
1
cucu
uu
!"
=
"
= !##
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Lecture 8
Energy-Momentum Transformation
[ ]
!!"
#$$%
&'
!#$& '
!"#$
%& '
=
()
*+,
-!"#$
%& ''''++'
!"
#$%
& '=
()
*
+,
-
!"
#
$%
& '+!"
#
$%
& '+'
!"#$
%& '
'=
.+.+.'=.
'
2
2
2
2
2
2
2
22
2
222
2
222
2
2
2
2
2
22
2
22
2
2
2
222
22
2
1
1
1
122
1
11
11)(
1
111
)()()(1
1)(
1
c
u
c
vu
c
v
c
vu
c
vuvvu
c
vuvuc
c
vuc
c
vu
c
vuvu
c
vuc
uuu
cc
u
x
xxxx
x
zyxx
zyx
$& '=( 21c
vux
uu )))
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Lecture 8
E andpTransformation-Cont
d
( )vumcmc
vucmcmE
xuu
xuu
!"=
&'( "==) )
0
2
0
2
2
0
2
0 1
***
***)( xvpE!= "
!!#
$$&
'(=
!"#$%& (
(!"
#$%
& (=
)=) )
2
2
00
2
02
0
1
1
c
cmvum
c
vu
vum
c
vu
ump
uxu
x
xxu
xux
***
**
*
!#$& '= 2cvEpx(
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Lecture 8
EandpTransformation-Cont
d
yu
x
yx
u
yuy
um
c
vu
um
c
vu
ump
0
2
02
0
1
1
!
!
!!
!
=
"#$%
&' ("#
$
%&
' (=
)=) )
yy pp =!
Similarly, we have zz pp =!
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Lecture 8
Summary
)(
2
x
zz
yy
xx
vpEE
pppp
c
vEpp
!="="
=
"
&'( !="
)
)
)(
2
x
zz
yy
xx
pvEE
pppp
c
Evpp
!+!=
!=!
=
%&' !
+!=
(
(
ct
c
E
rp
!
!!!
momentum-energy four vector:
$&
c
Ep,!
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Lecture 8
Lorentz Invariant Mass
As we have shown, the spacetime interval is Lorentz invariant:
22
22222
)()(
])()()[()()(
rtc
zyxtcs
!"!=
!+!+!"!=!
By analogy, the magnitude of momentum-energy four vector is
also Lorentz invariant:
2
2
pcE !"$%'
REST MASS!2
0 )( cm=