Modern Physics. Chapter 18 The Special Theory of relativity 狭义相对论基础 Chapter 19 The...
-
date post
20-Dec-2015 -
Category
Documents
-
view
274 -
download
6
Transcript of Modern Physics. Chapter 18 The Special Theory of relativity 狭义相对论基础 Chapter 19 The...
Chapter 18 The Special Theory of relativity
狭义相对论基础
Chapter 19 The Quantization of Light
光的量子性
Chapter 20 Quantum Theory of the Atoms
原子的量子理论
I. Three experimental milestones before the modern physics
1. The discovery of x-ray by Rontgen in 1895.
2. The discovery of radioactivity in 1896 by Becquerel.
With natural radioactive source (-particle), Ruthersford proposed the nuclear model of the atom in 1911.
x-ray incidents on graphite, the quantization of light was proved.(Compton scatter)
electron
3. The discovery of electron in 1897 by J.J.Thomson
II. Three theoretical milestones of the modern physics
1. The idea that electromagnetic radiation has particle characteristics as well as wave properties----Wave-Particle Duality of light.
Plank: Harmonic oscillators radiate energy only in quanta of energy h. (1900)
Einstein: the electromagnetic field consists of light quanta ( photon ). (1905)
Bohr: with the nuclear model, the electron and the particle nature of radiation to provide a model of the hydrogen atom (1913)
2.The theory of relativity proposed by Einstein in 1905.
Display a new thinking about space and time and mass-energy relationship.
3. The microscopic particles have the Wave-Particle Duality, too.
De Broglie hypothesis in 1924, Schrodinger equation in 1926, … Quantum Mechanics were established.
§18-2 The Postulates of Special Relativity 狭义相对论的基本假定
§18-4 Some Consequences of the Lorentz Transformation
洛仑兹变换的一些结果
§18-5 The Lorentz Transformation of Velocities 相对论速度变换式
§18-6 The Relativistic Dynamic theory 相对论动力学基础
§18-1 The Michelson-Morley experiment
§18-3 The Lorentz transformation 洛仑兹变换
§18-1 The Michelson-Morley experiment
The Michelson interferometer was used to look for ether.
P2M
1M
S 1l
2l
u(2)
(1)
O
u--the speed of the earth relative to the ether.
For the light beam (2) : O M2 O
uu
c
l
c
lt 22
2
)/1
1(
222
2
cc
l
u
For the light beam (1) : O M1 O
1M
1l21tc
1tu
O
A
B O
The actual path of beam (1) is O A O
In the OAB,
21
2121 )2
()2
( lt
ut
c
)/1
1(
222
11
cc
lt
u
Let l1 = l2 = l and using u<< c
)2
1(2
2
2
1 cc
lt
u
)1(2
2
2
2 cc
lt
u
12 ttt 32 / cluThe time difference of two light beams is
We get
The difference of optical lengths of two light beams is
tc 22 / clu
P2M
1M
S 1l
2l
u(2)
(1)
O
Let the interferometer rotates 900
P
2M
1M
S
1l
2lu
(2)
(1)O
12 tt 21 tt
900
12 ttt 32 / cluThe time difference of two light beams is
The difference of optical lengths tc The difference of optical lengths changes 2 because of the rotating.The number of interferometer fringes should shift
2
N2
22
c
lu
Taken c=3108 m/s, l=11m, =5.910-7 m, u= 3104 m/s (the speed of the earth moving around the sun)
4.0
The result of the experiment is
null result
i.e.
The ether does not exists and the speed of light relative to the earth is C.
I. I. Galileo transformation
x
y
z 0
K
ut'x
'y
'z'0
'KuP
§6-2 The postulates of special relativity§6-2 The postulates of special relativity
1. The eventIt happens in the space at sometime. --- recorded by the coordinates of space and time.When the observer locates in K-system, it is recorded with P(x, y, z, t)
When the observer locates in K-system, it is recorded with P(x, y, z, t )
2. 2. Galileo transformation
yy 'zz 'tt '
utxx '
'' utxx
'yy 'zz 'tt
Let 0 and 0 coincide when t=t=0
Then
Or
x
y
z 0
K
ut'x
'y
'z'0
'KuP
Relationship?(x, y, z, t) (x, y, z,
t )
uvv xx '
yy vv 'zz vv '
xx aa 'yy aa 'zz aa '
Transformations of velocities Transformations of velocities and accelerationsand accelerations
aa '
yy 'zz 'tt '
utxx '
II. The classical opinions about the time and spaceII. The classical opinions about the time and space
Absolute spaceAbsolute space ::
KK 212
212
212 )()()( zzyyxxr
KK
212
212
212 )''()''()''(' zzyyxxr
)()('' 1212 utxutxxx 12 xx
r
----The measurement of length is constant ----The measurement of length is constant in different coordinate system.in different coordinate system.
AbsoluteAbsolute time:time:
tt '
AbsoluteAbsolute mass:mass:
FF
'SoSo
tt '
mm
----The measurement of time interval is The measurement of time interval is constant in different coordinate system.constant in different coordinate system.
1.1. The principle of the relativityThe principle of the relativity :: the laws of the laws of physics are the same in all inertial reference physics are the same in all inertial reference frames.frames.------all inertial reference frames are equivalent.all inertial reference frames are equivalent.
22. . The principle of the constancy of the speed The principle of the constancy of the speed
of lightof light :: the speed of light the speed of light cc is the same for is the same for every inertial reference frame and is every inertial reference frame and is independent of any motion of the source.independent of any motion of the source.
§6-2 The postulates of special relativity§6-2 The postulates of special relativity
------there is no preferential frame in the universe.there is no preferential frame in the universe.
A event happens in the space at sometime, A event happens in the space at sometime,
§18-3 The Lorentz transformation
It is recorded with P(x, y, z, t) in K-system
and with P(x, y, z, t ) in K-system
x
y
z 0
K
ut'x
'y
'z'0
'KuP
Relationship(x, y, z, t) (x, y, z,
t )in relativity?
BtAxx
A event happens in the space at sometime, As A event happens in the space at sometime, As the space and time have the symmetry and the the space and time have the symmetry and the homogeneity, homogeneity, Their transformations should be linear.Their transformations should be linear.
LetLetyy
zz DtExt
A, B, E, D are constants.
Using the principle of the relativity:
Observing in K-system,
O locates in x=ut at time t.
x =0 BtAxx
,AuB Inversely, observing in K-system,
O locates in x =-ut at time t.
x =0
)( utxAx
)( utxAx
-ut 0
=Ex+DtAD
Using the principle of the constancy of the speed of light:Assume a light beam is emitted in the origin of K and K at time t =t = 0,
The signal of light arrives x =ct in K-system at time t and x=ct in K-system at the time t .
)( utxAx
AtExt
ct ct2c
AuE
SoSo )( utxAx yy
zz )(
2x
c
utAt
Find A=?
Meanwhile, the signal of light arrives a point on y axis with (y =ct, x=0) in K-system at time t.
And this point is measured with (x, y ) in K-system at time t .
i.e., in K-system :
x= 0
y=ct
in K-system :
2222 tcyx
Lorentz transformation
221'
cu
utxx
yy 'zz '
22
2
1'
cu
xcu
tt
its inverse transformation
221
''
cu
utxx
'yy 'zz
22
2
1
''
cu
xc
ut
t
L-transformations have no scene if L-transformations have no scene if v v >c>c
---- ---- cc is the ultimate speed of an object.is the ultimate speed of an object.
The space and time relate to each other in The space and time relate to each other in L-transformations.L-transformations.
NotesNotes
WhenWhen u u<<c<<c,, 221 cu 1utxx '
tt '--L-transformation--L-transformation Galileo transformation
If If two eventstwo events happen in the space at sometime, happen in the space at sometime,
they are recorded with P1(x1, y1, z1, t1) and P2(x2, y2, z2, t2) in K-system,
P1(x1, y1, z1, t1 ) and P2(x2, y2, z2, t2 ) in K-system,
then, then, 12 xxx
22
2
1'
cu
xcu
tt
221 cu
tuxx
12 ttt
12 xxx
12 ttt
221'
cu
udtdxdx
22
2
1'
cu
cudxdtdt
'
''
dt
dxvx 2cudxdt
udtdx
21 cuv
uv
x
x
dydy '
dzdz '
§18-5 The Lorentz Transformation of Velocities
'
''
dt
dzvz
2
22
1
1
cuv
cuv
x
z
Similarly, Similarly,
'
''
dt
dyv y
2
221
cudxdt
cudy
2
22
1
1
cuv
cuv
x
y
I. The relativity of lengthI. The relativity of length
§§18-4 Some Consequences of the Lorentz Transformation
The proper length (The proper length ( 固有长度固有长度 ) ) LL00 :: the lengtthe lengt
h measured by the observer who is at rest relah measured by the observer who is at rest relative to the object.tive to the object.
x
y
0
K 1x
'' 120 xxL
u
2x 'x
'y
'0
'K
InIn KK-system-system ::?
A rod is at rest in KK-system,-system,
the spaceship is the spaceship is movingmoving relative to relative to KK-system-system,,
so we must measured so we must measured xx1 1 and and xx22 simultaneouslysimultaneously
in in KK-system,-system,
i.e.,i.e.,at timeat time 21 tt tmeasuremeasure
12 xxL
x
y
0
K
u
1x 2x
'x
'y
'0
'K
'' 120 xxL
22
1
22
2
11 cu
utx
cu
utx
22
12
1 cu
xx
221 cu
L
220 1 cuLL 0L
----length contraction (length contraction ( 长度收缩长度收缩 ))
NotesNotes ::The measurement of length is not absolute The measurement of length is not absolute
and the proper length and the proper length LL00 measured by an measured by an
observer who is at rest relative to the object is observer who is at rest relative to the object is the largest.the largest.
When the object is moving relative to the obsWhen the object is moving relative to the observer, the coordinates of the two side must be erver, the coordinates of the two side must be measured measured simultaneouslysimultaneously..
Length contraction happens only on the Length contraction happens only on the direction of the object moving.direction of the object moving.
All inertial reference frames are equivalent.All inertial reference frames are equivalent.
2. The relativity of time( the time dilation effect)2. The relativity of time( the time dilation effect) Proper time interval (Proper time interval ( 固有时间固有时间 ) ) tt00 :: the timthe tim
e interval measured by the observer who is at e interval measured by the observer who is at rest relative to the rest relative to the two events happening at thtwo events happening at the same point of the spacee same point of the space..
x
y
0
K
0x
flashflash10 , txEvent 1Event 1
Event 2Event 2 20 , tx
120 ttt
In K-system:
x
y
0
K
0x
u
'x
'y
'0
'K
12 ttt 22
201
22
202
11 cu
cuxt
cu
cuxt
22
0
1 cu
t
0t
--time dilation --time dilation ( ( 时间膨胀时间膨胀 ))
In K -system:
NotesNotes ::
tt00 is the smallest interval between two eventis the smallest interval between two event
s that any observer can measure.s that any observer can measure.A moving clock relative to the observer run slA moving clock relative to the observer run sl
owly than a static clock relative to the observowly than a static clock relative to the observer----moving clock run slow er----moving clock run slow (( 动钟变慢动钟变慢 ))
b'x
'y
'0
'Kthe flash travels one the flash travels one period in period in KK-system, -system,
c
bt
2' 0t
Flash clock
tu
bl l
bl l
x
y
0
Ku
'x
'y
'0
'K
22
0
1 cu
tt
c
lt
2 22 )
2(
2 tub
c
Observing in Observing in KK-system,-system,the flash-system is movingthe flash-system is moving
moving clock run moving clock run slow slow
u
'x
'y
'0
'K3.The relativity of 3.The relativity of “simultaneity”“simultaneity”
x
y
0
K
1x 2x
(“(“ 同时”的相对性同时”的相对性))Assume Assume two eventstwo events
happen,happen,
they are measured by they are measured by an observer locating in an observer locating in KK–system, –system,
12 ttt 0their positions are their positions are xx11, , xx2 2 andand
simultaneoussimultaneous
22
211
22
222
1
1
cu
cuxt
cu
cuxt
12' ttt
22
221
1
/)(
cu
cuxx
0
In In KK –system, –system, u
'x
'y
'0
'K
1x 2x x
y
0
K
is not is not simultaneous simultaneous observing in observing in KK –system–system
The sequence of the eThe sequence of the events happening:vents happening:
x
y
0
K
1x 2x
u
'x
'y
'0
'K
----the event locating on the event locating on xx22 happens happens
first first observing in observing in KK –system–system..
22
221
1
/)(
cu
cuxx
0
12 tt
NotesNotes ::simultaneity is not an absolute concept but a simultaneity is not an absolute concept but a
relative one, depending on the state of motion relative one, depending on the state of motion of the observer.of the observer.
12 tt't
The “start-end” sequence of the event is The “start-end” sequence of the event is absolute in any inertial reference frame. absolute in any inertial reference frame.
22
2
1'
cu
xc
ut
t
22
2
1
)1(
cu
c
uvt
cu cv as
't tand and havehave same signsame sign
-- -- The “start-end” sequence of the event The “start-end” sequence of the event does not change. does not change.
t
xv
[[ExampleExample] A observer ] A observer AA saw that two events saw that two events locating on locating on xx axis happen simultaneously. Their axis happen simultaneously. Their distance is distance is 4m4m. Another observer . Another observer BB saw that the saw that the distance between the two events is distance between the two events is 5m5m. . Do the Do the two events happen simultaneously relative to the two events happen simultaneously relative to the observer observer BB? ? how much ishow much is the time interval the time interval measured by measured by BB ? ? how much is how much is the relative the relative speed between speed between A A andand B B =? =?
SolutionSolution :: let let AA locates on locates on K-K-systemsystem and and B B lolocates on cates on KK-system-system.. BB is moving relative t is moving relative to o AA with the speed with the speed uu along along x x axis.axis.
m4x m5'x0tthenthen
§18-6§18-6 The Relativistic Dynamic theory
1. Relativistic mass1. Relativistic mass
When an object with static mass When an object with static mass mm00 is moving is moving
with the speed with the speed vv ,, its moving mass isits moving mass is
22
0
1)(
cv
mvmm
DeduceAssume there are two same particles A and B ,
and relative speed is v
their static mass is m0 respectively,
A BvIf A and B collide in absolute
non-elastic,
the momentum of A and B is conservative.
Let B is K-system,)1()( 0 cVmmmv
Vc--- The speed of A+B’s center of mass relative to K-system
A is K’-system :
)2()( 0 cVmmmv
Vc’--- The speed of A+B’s center of mass relative to K’-systemCombining (1) and (2), we get
)3(cc VV And using Lorentz transformation of velocity
)4(1
2
c
vVvV
Vc
cc
A Bv
DiscussionDiscussion
If If vv<<<<cc , , )(vm 0m
If vv cc, , )(vm a 0------The speed of an object cannot exceedThe speed of an object cannot exceed cc
2. Relativistic momentum and dynamics equation2. Relativistic momentum and dynamics equation
vmp
22
0
1 cv
vm
MomentumMomentum
3. Relativistic energy3. Relativistic energy
An object has a displacement under the An object has a displacement under the action of a force . action of a force . F
rd
rdFdEk
rd
dt
vmd
)(
vvmd )(
dmv 2 mvdv
kk dEE mvdvdmv 2 ?
The increment of its kinetic energy isThe increment of its kinetic energy is
As
2
2
0
1c
v
mm
We get22
02222 cmvmcm
Making a differential on the two side of equation
mvdvdmvdmc 22
kk dEE m
mdmc
0
2
20
2 cmmc Kinetic energyKinetic energy
Total energyTotal energy Rest energyRest energy
DiscussionDiscussion
Rest energy contains the total inteRest energy contains the total inte
rnal kinetic energies of all particles moving irnal kinetic energies of all particles moving i
n the object n the object + + the total internal potential enthe total internal potential en
ergiesergies
200 cmE
--Internal (intrinsic)energy of the object--Internal (intrinsic)energy of the object
--micro-energy--micro-energy
Kinetic energy is the macro-eneKinetic energy is the macro-ene
rgy when the object has a mechanical motiorgy when the object has a mechanical motio
n as an entirety.n as an entirety.
0EEE k
cv ,,IfIf
]1)(2
11[ 22
0 c
vcm
202
1vm 2
0cm
--------Large part of energy storesLarge part of energy stores in in the objectthe object
20
2 cmmcE k )11
1(
22
20
cvcm
Total energyTotal energy 2mcE kEE 0
Mass-energy conservation lawMass-energy conservation law
In an isolated system of particles, the In an isolated system of particles, the total total energyenergy remains constant. remains constant.
i
ii
i cmE 2=constant=constant
the the total mass of the systemtotal mass of the system is conservative is conservative
i
im =constant=constant
The conservation of The conservation of total energytotal energy is is equivalent to the conservation of equivalent to the conservation of total mass.total mass.
Rest energy and kinetic energy can convert eRest energy and kinetic energy can convert each other.ach other.
ExampleExample: nuclear fission: nuclear fission
Before fission, the object has Before fission, the object has 2
00 cMEE )0( 0 kE
After fission, the object break into two particles. After fission, the object break into two particles. Their total energy is Their total energy is
22
21 cmcmE
22
2012
10 kk EcmEcm
Total energy is conservative:Total energy is conservative:
Total mass is conservative:Total mass is conservative:210 mmM
The loss of rest mass :The loss of rest mass :
201000 mmMm
The loss of rest energy :The loss of rest energy :2
0cm 21 kk EE
The increment of kinetic energy .The increment of kinetic energy .
22
20
1 cv
cmE
22
0
1 cv
vmp
Eliminating Eliminating v , v , we get we get
22420
2 cpcmE
2220 cpE
pc
0E
E 动质动质能三能三角形角形
4. The relationship between the total energy and 4. The relationship between the total energy and momentummomentum
5. Photon5. Photon
Total (moving) massTotal (moving) mass
22
0
1 cv
mm
Rest mass Rest mass mm00=0=0
momentummomentum
2c
Em
c
Ep
c
h
h
2c
h
22420
2 cpcmE
[[ExampleExample] A particle with rest mass ] A particle with rest mass mmoo is moving is moving
at the speed at the speed vvoo=0.4=0.4cc. If the particle is accelerat. If the particle is accelerat
ed till its final momentum = ed till its final momentum = 10 10 initial momentinitial momentum. What is the ratio of its final speed and initum. What is the ratio of its final speed and initial speed?ial speed?
SolutionSolution initial momentum is initial momentum is
220
000
1 cv
vmp
2
0
4.01
4.0
cmcm044.0
010 pp cm04.4Final momentumFinal momentum
[[ExampleExample] Two particles have rest mass ] Two particles have rest mass mmoo
respectively. One particle is at rest and another respectively. One particle is at rest and another is moving with the speed is moving with the speed v v =0.8c=0.8c. They collide . They collide and stick together. Find the rest mass of the and stick together. Find the rest mass of the compound particle.compound particle.
解解:: Assume the mass of the compound particlAssume the mass of the compound particle is e is MM and speed is and speed is VV after colliding after colliding
Mmm 0
The mass of the system is conservative The mass of the system is conservative before and after colliding:before and after colliding:
The momentum is conservative:The momentum is conservative:
MVmv