phys541 量子力學 (i)(ii) 清華大學物理系牟中瑜. mou/teach.html...
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PHYS541量子力學 (I)(II)
清華大學物理系牟中瑜
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s
http://www.phys.nthu.edu.tw/~mou/teach.html
http://www.phys.nthu.edu.tw/~mou/teach/11Fall_QuantumMech.html
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http://www.phys.nthu.edu.tw/~mou/teach/11Fall_QuantumMech.html
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Quantum mechanics – New way that was developed at the beginning of the last century to interpret & predict microscopic objects such as atoms, electrons, ..
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Purpose of this course Learn to calculate and think in quantum mechanical way
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I think that it is fair to say that no one understands the quantum theory…
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Conceptual problem will not be fully discussed in this course. In fact, as Richard Feynman remarked:
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Two tracks of discoveriesSpectroscopy
(discrete)
Theory of H Atom (Bohr)
Matrix Mechanics(Heisenberg)
h
Photoelectric Effect (Einstein)
Matter wave(De Broglie)
Wave Mechanics (Schrodinger)
Equivalent (shown by Schrodinger)
(based on the general formalism of Dirac))
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Third Formulation
Richard FeynmanPath Integral
Equivalent to the above two formulations but may be more general
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Track 2: 粒子與波的爭議惠更斯 : 光是波動
牛頓 : 光是粒子虎克 : 光是波動
馬克士威 : 光是波動楊格 : 光是波動
卜朗克、愛因思坦、康卜吞 :光是粒子
..
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光的馬克思威耳理論
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光的波動性
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光的粒子性Pierre Gassendi 與牛頓
光的反射:粒子的彈性碰撞
光的折射: 1
2
1y
2y
1
1
1 2
1 2 2
2
21 1 2
If
sin
s
i
n
/
y y
n
n
n
n c n
及
與不合
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幅射與黑體幅射 : 物體皆會幅射電磁波
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Max Planck
Bk T古典平均每一波動能量為
2 給定一個頻率 約有 個不同方向的波
Expt & Planck
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E
( 2.71828..)
B
E
k Te
e
一個平衡系統中
能量為 之機會
/ 2 / 3 /
/
,
2
Plank:
1
3 B B B
B
h k T h k T h k T
h k T
h
h e h e h e
h
e
光波的能量不連續
基本單位為
平均能量為
h = 卜朗克常數 =6.62610-34 joule-sec
Planck’s consideration is based on statistical mechanics not dynamics
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Quantum region
3 3=N (kinetic energy) N (potential energy)=3N
2 2
3N
B B B
B
U k T k T k T
dUC k
dT
Solid State Version : C versus T
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光電效應 :
金屬
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電子放光的波動模式
電子所經歷之電場
電子
x
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波動的期待
電子所經歷之電場
電子
xE
• E 愈大 ( 光強度愈大 ) ,愈容易搖出電子,電流愈大。與波長、頻率無關。
• 任何頻率的光皆可產生光電子• 需要有搖動累積能量的時間
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光電效應實驗
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Einstein: E=nh is dynamical!
, h 光由光量子組成 每一個光量子能量為
21-
2m h 2
0 max 0
1- min /
2eV h W m f W h
W
h
W 功函數
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Compton effect:
: = 波動
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Compton :粒子碰撞的結果
光量子與電子之彈性碰撞 : 碰撞後光子動量改變 , 則波長改變
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粒子碰撞
ei f
i f e
2 2e
f
2e
e
2
h hcos p cos
hsi n p s
h =h E
E p c (mc
n
)
i
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電路元件的基本單元電子是波動還是粒子 ?德布羅依博士論文:電子有波動性
1924 德布羅依加入爭辯
光是波動還是粒子 ?
hp
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什麼是波動? 什麼是粒子 ?
粒子:(1) 一顆一顆,不連續(2) 有固定軌跡
波動:(1) 干涉(2) 繞射
皆滿足能量與動量守恆
需要更明確直接的實驗來說明 !
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電子源
粒子的期待
如果是粒子
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如果是波動
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電子源
“ 電子版”的楊氏干涉
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d
dsin = m 建設性干涉
L
yd
L
Ly
d
dsin = m dsin = (m+1/2) 破壞性干涉
定量分析
sin tany
L
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Ly
d 7mm
1L m
1 410 10d mm m 700nm 0.17nm
y 1.7 m
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The Feynman Lectures on Physics (III) p. 1-4~1-5
…This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossible small scale … We are doing a “thought experiment”…
參考值 : Davisson and Germer 之電子繞射實驗 電子波長為 0.165nm(1.65 Å, 50eV)
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Tonomura et al.American Journal of Physics57, 117(1989)
= 0.054Å (50kV), Va = 10V
a = 0.5m, b = 5mm
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中子
Reviews of Modern Physics 60, 1067 (1988)
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氦
Physics Review Letters 66, 2689 (1991)
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http://www.quantum.univie.ac.at/research/c60/index.html
大分子之物質波
C60
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Nature 409, 304(2001)
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(1) Diffration grating is SiNx grating (period 100 nm) with width 0.1 m.(2) C60 is thermal ionized by a laser. The ions are then accelerated and directed towards a conversion electrode. The ejected electrons are subsequently counted by a Channeltron electron multiplier.
Nature 401,680 (1999)
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Other atoms:Na, Phys. Rev. 66, 2693 (1991)
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生物分子
3D structure of tetraphenylporphyrin C44H30N4(TPP)
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L Hackermuller et al. Phys. Rev. Lett. 91 090408, (2003)
3D structure of the fluorofullerene C60F48
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Science 275, 637 (1997)
106 鈉原子的干涉
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結論
(i) 出現之次數 ~ 粒子出現的機會
(ii) 當有許多互相排斥之選擇時 P ( 總機會 ) P1+P2+P3+ …
不同於粒子的行為 : P = P1+P2+P3+ …
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如何描述 ?
(i) 光強度 (I) ~ |E|2
(ii) 當有許多選擇時 I (total intensity) ~ |E1+E2|2 |E1|2+|E2|2
因而跳脫了 P (total probability) = P1+P2
來自電磁波之經驗 :
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Max Born 的機率解釋 :
(i) 發生 ( 出現 ) 機會 ~ | |2
(ii) 當有許多選擇時 , 每一選擇以 i 表示 總機會 ~ |1+ 2+ …. |2 ( 疊加原理 )
(x,y,z,t) = 物質波波函數(complex)
E(x,y,z,t) = 電場
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古典與量子的差異 :|1+ 2|2 – (|1|2+ |2|2)= 12
*+ 1*2 = 干涉項
r1
r2
1
1
2
2( )2 21 2
1 2
2
1
| | |1 |
2 2cos ( )
ik r
ikr
ikr
re
k r
e
r
e
1 2 1 2
2( ) ( ) 2k r r r r
路程差
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古典 : |1|2+ |2|2 量子 : |1+ 2|2
電子源
1
2
因此電子不是通過 1 或通過 2,而是可以“同時通過 1 與 2”
全新的觀念
參考書 : 原子中的幽靈 ( 貓頭鷹書房 )
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五大量子效應 (to be covered)
干涉量子化穿隧效應自旋量子心電感應 (entanglement)
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量子心電感應 (Entanglement/ 糾纏 )
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焦平面
糾纏雙光子實驗
a
b
'b
'a
光子 1 光子 2
?
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Dopfer, B., 1998Zeilinger, Rev. Mod. Phys. S288, (1999)
焦平面上 Detector 之距離
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55
Spooky action: 鬼魅般的行為
Einstein: Spooky action at a distance!
機率的本質使資訊的傳播無法超過光速
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Feynman: I think that it is fair to say that no one understands the quantum theory…
s
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波動的個數
xy
z
2 24
頻頻為 之可能方向數
c k