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科目:近代物理
編著:秦立原
本power point教學檔案內容,乃教師本人根據
Arthur Beiser之Concepts of modern Physics
原著演繹並解讀而成。請尊重智財權,勿任意下載流通。
Special Relativity
• Sec. 1.1: All motion is relative; the speed of light in free space is the
same for all observers.
remark:
(1)The above two statements have no dependence, the first is not the
cause of the second and the second is not the cause of the first.
(2)All motion is relative means that, when we say something A is in
motion, the motion must relative to something B). And B is also in
motion to A ,this is just the thing the first statement want to express.
(3)Observers in the second statement represents inertial frames of
reference. An inertial frame of reference is one which Newton’s first
law of motion holds. Any frame of reference that moves at constant
velocity relative to an inertial frame is itself an inertial frame
(4)The word ‘free space’ in the second statement is very important. The
speed of light in free space is 2.998×108m/s for all inertial frames of
reference.
Postulates of Special Relativity
• Postulate 1:The laws of physics are the same in all inertial frames of
reference.
Remark: So called the laws of physics includes: (p15., Eisberg)
(1)Electromagnetic phenomena (including the fact that the propagation
velocity of light is equal to the constant value c)
(2)The laws of mechanics:
Fig.1.6 discuss this postulate. In that fig., the speed of a spacecraft
relative the earth is assumed to be v and v is greater than c (light
speed in free space). A man in the spacecraft switches on a flashlight
to the front of the spacecraft and he would see the flashlight
illuminates the front wall of the craft.
However, the observer on the earth would see the flashlight
illuminates the back wall of the craft, because v>c in the
earth frame of reference.
Now, the flashlight is seen to illuminates different parts of the craft by
different frames.
It is just the different results which conflict with the postulate one.
So we know the assumption that v, the speed of any spacecraft must not
greater than the constant c. And we can conclude that the speed of
nothing can greater than the constant c
• Postulate 2: The speed of light in free spacehas the same valuein all
inertial frames of reference.
Unlike postulate 1which is based on pure thinking and life
experience, postulate 2 is based on the resultes of many experiments.
Fig1.1illustrates postulate 2. Jack is in the earth, Lee in a spacecraft.
When the craft pass him with v=(2/3)c, Jack turn on a searchinhlight.
Jack will surely find the speed of the searchinglight is c.However,
Lee will find the speed of the searchinglight is also c, but not (1-
2/3)c.
Particle properties of waves
• In classical physics,particles and waves are separate components of
physical reality. The mechanics of particles and the optics of waves
are traditionally independent disciplines. They both have thtir own
serious experiments and principles correspond those experiments. • However, we find that a moving particle,such as an electron, can be
seen as a wave.
• Simiarly, under some cases,electromagnetic waves behave as if they
consists of streams of particles. Then we can see an EM wave as a
particle.
• Together with special relativity,the wave-particle duality is the
central of modern physics.
• In 1864, Maxwell suggested that accelerated charges generate linked electric and magnetic disturbances that can
travel infinitely through space.
• If the above mentioned charges oscillate periodically, disturbnaces will becomes waves, and E,B,v will prependicular to each other, where v is the propergation
velocity vector of EM waves.
• Before the suggestion was proposed by Maxwll, people only knew the Farady’s induction law: A changing magnetic
field can induce a current in a wire loop.
• Maxwell proposed the converse: a changing electric field
has a changing magnetic field associated with it.
• The product of electric permittivity and magnetic permeability is
derived to be equal to the inverse of c2, it can’t be just an accident.
So Maxwell again suggested that light consists of EM waves.
• During Maxwell’s lifetime, No experiments found the existence of
EM waves.
• In 1888, Heinrich Hertz’s experiment showed the existence of EM
waves. He found the EM waves could be reflected,refracted, and
diffracted.
Blackbody radiations
• When discussing blackbody radiations, radiations are not reflections.
They are the nature ability of matters, the higher the surrounding
temperature is, the stronger the radiations are.
• Why did scientist be interested in blackbody? The answer is the
reflection ratios of different matters are also different. It effects the
detecting of radiations. If scientist could exclude the existence of
reflections, they would be able to see all the collected light as pure
radiations without any reflections. So that scientist can concentrate
their attention on the relation between the intensity of radiations and
temperature.
• The ability of a body to radiate is proportional to its ability to absorb
radiation.
• In blackbody spectra. The spectral distribution of energy in the
radiation depends only on the temperature. The higher the
temperature, the greater the amount of radiation and the higher the
frequency at which the maximum emission occurs.
• Rayleigh and Jeans considered the radiation inside a cavity of
absolute temperature T whose walls are perfect reflectors to be a
series of standing em waves. They also combined the formulas of
the standing waves density and classical average energy per standing
wave.
• So Rayleigh and Jeans derived (2.3)
Plank’s treatment for blackbody
radiation In the cavity of blackbody m odel, each standing w ave represents an EM w ave
and em itted from an oscillator. T he ene rgy of each oscillator is . In the
cavity, the value of is not uniq, just as
nnh
the case of the standing w ave generated
from a vibrating string ,w hose length =L and tw o ensd is fixed.
T he focus of P lank's treatm ent is not th e distribution of , because ( ) has
been derived to be
v G v dv
2 3 8 / .
T he focus of P lank's treatm ent is the distribution of , consequently
the distribution of . T he w eight of is exp( / ) for a fixed . So
the the avergy energy of an oscillator i
n
n n
v dv c
nhv
n kT v
s
[ exp( / )] /[ exp( / )]
[ exp( / )] /[ exp( / )]
= /[exp( / ) 1] (2.6)
n n n nkT kT
nhv nhv kT nhv kT
hv hv kT
• (2.6) is also the average energy of the standing waves with the fixed
frequency ν in the cavity. For those standing waves with greater
ν. This average energy will be smaller so that cancel the effect of
G(ν) for the greater ν .
• Example (2.1) talk about the evidence of quantum effect. The energy of
a quantum, h ν, is 10-29 times of n h ν ,0.04J, the total energy of the
fork. 10-29 times is so small that we say the evidence of the quantum
effect is very small (But we can’t say there is no quantum effect ).
• If we want to increase the evidence of quantum effect, we should
decrease the value of n very rapidly. If its value (1029) can be decreased to
smaller than 10, we should be able to say the quantum effect is evident.
But the strike force to the fork cannot be decreased to so small a value
(Remember that the value of the fork is 0.04J, It’s uneasy to make it
become 0.04× 10-28 ~ 0.04× 10-29 )
例(3.1a)中golf ball 非oscillator,故欲論其能量量化
明顯與否時,必須先令其能量可以量化。故須令golf
ball 侷限於箱中運動。而此例中ball 活動空間並未受
侷限,故只能討論其波動行為之明顯性。此例中之golf
ball 雖波長太小而不易發現其波動行為;但因與例b中
之electron均具明確之波長故△λ=0, s.t. △p=0,使
△x=∞(由測不準原理) 。故在運動方向上各處均可出
現,而並未被侷限住(例3.5中之marble就被侷限住了,故△p≠0,s.t. △λ≠0且能量被量化)
對應原理
• 量子數越大時,量子物理越接
近古典物理。
• 一個繞行圓形軌道的電子所輻射出來的電磁波頻率會等於旋轉的頻率和旋轉頻率之諧波頻率(也就是旋轉頻率的整數倍)。
– 旋轉頻率
– 光子頻率
– 軌道穩定的條件 2
nhm r
5.2 波動方程式 The Wave
Equation
( )i t xy Ae
2 2
2 2 2
1y y
x t
xy F t
波動方程式可有許多種類的解,包含了複數型式。
波動方程式
偏微分
cos sinx x
y A t iA t
-15•(1) = 500 , (
( )
_____ (b)
( : 4 .136 10 )pc keV
keV c
h eV s
-12
寫 出 計 算 過 程 並 將 答 案 填 入 空 格 中
已 知 ㄧ 粒 子 之 靜 止 能 量 及 其 值 均 求 a)此 粒 子 之 波 長 =___ 10 m
此 粒 子 之 動 能 =_________ 此 粒 子 之 相 速 度 =_____c (d)此 粒 子 之 群 速 度 =_____c
(2)某 粒 子 被 限 於 x=0至 x=L之 無 限 堅 硬 一 維 箱 中 ,(a)x=L處 , 此 粒 子 之 之 波 函 數 為
提 示
(b)
n
2 = sin n=1
____ (c) =_____
,
n x
L L
則 對 之 狀
況 而 言 ,粒 子 出 現 在 處 之 機 率 最 大 , 前 述 之 機 率 最 大 值
(3)設 一 限 制 於 x軸 上 之 粒 子 ,在 x=0與 x=1之 間 的 波 函 數 為 =ax;而 在 其 他 位 置 則 為 =0.(a)求 此 粒 子 在 x=0至 x=0.5間 出 現 的
機 率 =______? (b)求 粒 子 位 置 之 期 望 值 <x>=______
(4)物 質 波 之 雙 狹 縫 繞 射 實 驗 中 若 將 狹 縫 1,2分 別 打 開 所 得 波 函 數 分
x=
1 2, 別 為
密 度 =
及 則 將 兩 狹 縫 同 時 打 開 時 ,螢 幕 上 之 機 率
_____________
近代物理期末考題及題解