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。

Modern Physics

• Special relativity

• Particle characteristics of wave

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)

• 原子核的尺寸

– 最靠近的距離

Chin:(1)此值係指動能全耗盡而變為電位能時之距離

(2)拉塞福之理論用至 推導Au原子核之尺寸,其值雖為

實際值之五倍但於數量級上之評估已堪稱卓越也

原子的行星模型及其為何失敗

–電子速度

–氫原子的總能量

原子中的電子波動

–軌道上的電子波長

–軌道穩定條件

–波耳原子的軌道半徑

–波耳半徑

1, 2, 3 ,n2 n

n r

11

0 15.292 10 ma r

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

量子力學 Quantum Mechanics

Chin:粒子存在整個空間中之總機率為1，此可將波函數予以正規化。

Chin:波函數絕對值之平方為機率密度而不為機率。

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

5.3 薛丁格方程式：時間相依型 Schrödinger’s Equation: Time-Dependent Form

2

( , )2

pE U x t

m

一維的時間相依型薛丁格方程式如下:

-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

(1之c,d)