Modern PhysicsModern Physics

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.

Chapter 18The special theory of

relativity

§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

221

1

cuA

We getWe get ：：

2222 tcyx =A(0-ut) =y=ct =A(t-0)

)( utxAx yy

)(2

xc

utAt

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

21 cuv

uvv

x

xx

2

22

1

1

cuv

cuvv

x

yy

2

22

1

1

cuv

cuvv

x

zz

uu<<c<<cuvv xx '

yy vv '

zz vv '

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

221'

cu

tuxx

221

45

cu

We get We get cu 6.0

22

2

1'

cu

xc

u

t

221

46.0

cuc

s810

§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

Combining (3) and (4) , we get

512

22

v

cc

v

cVc

Substituting (5) into(1), we get

2

2

0

1c

v

mm

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

Dynamics equationDynamics equation

dt

vmdF

)(

)1

(22

0

cv

vm

dt

d

dt

dmv

dt

vdm

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

22

0

1 cv

vmp

AsAs getget cv 975.0

4.0

975.0

0

v

v87.2

[[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

22

0

1 cv

mm

2

0

8.01

m

6.00m

mmM 0 03

8m

M

mvV c5.0

220 1 cVMM 2

0 5.013

8 m

031.2 m