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Fundamental Principles
of Quantum Mechanics
量子力學的基本精神
量子力學與人生一樣 –都充滿了不確定性
簡介
•量子化的觀念
Planck ( 1900 )簡諧振盪體 ( SHO ) 的能量量子化 ( 不連續性變化 )
Einstein ( 1905 )電磁波量子化光子 ( photon )
Bohr ( 1913 )原子的能量量子化 ( 不連續性變化 )半古典原子模型
•進一步介紹兩個正式建立量子力學 ( Quantum Mechanics ) 的基本觀念
de Broglie ( 1925 ) de Broglie’s hypothesis ( de Broglie 假設 )
Heisenberg ( 1927 ) Uncertainty Principle ( 測不準原理 )
de Broglie 假設 ( de Broglie’s Hypothesis –1925 )
•The motion of a particle is governed by the wave propagation
properties of a “pilot”wave called matter wave ( 物質波 )
= h / p : de Broglie wavelength
p : particle momentum
= E / h E : particle energy
例 : 一顆子彈 , m = 0.1 kg , v = 103 m/sec
= h / p = 6.63 x 10-34 / ( 0.1 x 103 )
= 6.63 x 10-36 m = 6.63 x 10-26 Å
•要觀察到一個粒子的波動特性 ( wave nature of a material particle )
de Broglie wavelength ( ) Å
Prince Louis-Victorde Broglie
( 1892 –1987 )
電子繞射實驗 ( Davisson-Germer Experiment )
例 : 一個電子 , m = 9.1 x 10-31 kg , v = 6 x 106 m/sec
= h / p = 6.63 x 10-34 / ( 9.1 x 10-31 x 6 x 106 )
= 1.2 x 10-10 m = 1.2 Å
•晶體中原子間的距離約為Å 範圍 ,
與電子的物質波波長大約相同
應該可以造成繞射現象
Davisson-Germer Experiment
( 1927 )
•from = h / p { need p } { for p need m }
V
•diffraction peak when n= 2d sin
d
•first, do x-ray diffraction ( x-ray = 1.65 Å )
change ( 90o 0o ) , and observe first peak at = 50o ( n = 1 )
= 65o ( 2+ = 180o )
inter-atomic plane distance, d = 0.91 Å
•fix = 50o and = 65o
change e- energy ( i.e. change V ) and measure I
•use de Broglie’s equation
= h/p = h / ( 2m e V )1/2
( K.E. = p2 / 2m = e •V )
V = 54 V = 1.67 Å , almost the same wavelength of the x-ray
電子具有波動的特性
V
I
54 V
V
n= 2dsin
n = 2dsin/ nn = h / ( 2m e Vn )1/2
V1 = 54 V1
V2 = 216 V2 = 1/2V3 = 486 V3 = 1/3
•e- reach Ni with Ek = e •V = ½ mv2 = p2 / 2m
•fix V = 54 V , measure intensity of the scattered e- beam ( I )
for different
物質波的本質 ( Nature of the Matter Wave )
電子束的雙狹縫干涉實驗 ( electron version of double-slit experiment )
OR
物質波的本質 ( Nature of the Matter Wave )
電子束的雙狹縫干涉實驗 ( electron version of double-slit experiment )
•物質波或粒子波所代表的是粒子出現機率的機率波
( waves of probability )
•wave magnitude indication of the probability that the particle
will be found at that point
–哥本哈根詮釋 Copenhagen Interpretation
•波幅 ( wave magnitude ) 越大
粒子出現的機率越大
• in particle model :
I ( intensity ) N , density of the particle
in wave model :
I ( amplitude of the wave )2
N ( amplitude of the wave )2
“GOD does not play dice with the universe ! “
- Einstein
這一個電子要落在那裡 ?
電子束的雙狹縫干涉實驗 ( electron version of double-slit experiment )
vx
理論上波的雙狹縫干涉條紋
是不是這一個粒子的質量會散開 ,並且分佈成這一種分佈圖形 ?
單電子發射的雙狹縫干涉實驗( double-slit experiment with series of single electron emissions )
上帝的確是在玩擲骰子的遊戲 !
100 electrons 3000 electrons 70,000 electrons
•量子力學主要在幹什麼 ?
尋找或計算一個物理系統 ( 單粒子或多粒子系統 ) 的波函數
•let the particle wave be represented by ( r, t )
( r,t ) : wave function ( 波函數 ) , r : position
•The probability that the particle will be found in volume dV :
P( r,t ) dV = 2 dV , P( r,t ) = 2 : probability density
2 dV = 1 ( normalization )–
測不準原理 ( Heisenberg Uncertainty Principle )
•electron single-slit
diffraction
as e- wave goes through slit
uncertainty in lateral
position = d = x
uncertainty in momentum
in x-direction = px
px > p sin , sin= /d = /x
p = h / px > ( h/) sin= ( h/) ( /x ) px •x > h
d locate the particle more precisely
greater uncertainty in momentum
d ~ Å
e-
py
Intensity2
y
x
ppy
px
•Heisenberg Uncertainty Principle :
one can not determine the exact value of x ( position ) and px
( momentum ) of a particle simultaneously
px •x h , h = h / 2
例 : try to predict the motion of moon around the earth
if need to have x = 10-6 m
mmoon = 6 x 1022 kg , vmoon = 103 m/sec
px > h / x 10-34 J sec / 10-6 m = 10-28 kg m/sec
vx = px / m > 10-28 / 6x1022 10-50 ( m/sec )
( compare to v = 103 m/sec )
Werner Heisenberg( 1901 ~ 1976 )
例 : consider e- in H atom, ro 0.5 x 10-10 m ( 0.5 Å )
let x ~ 10-10 m ( 1 Å )
px > h / x = 10-34 / 10-10 = 10-24 ( kg•m / sec )
Ek = 13.6 eV = 2.18 x 10-18 J
p = ( 2m Ek )1/2 = ( 2 x 9.1 x 10-31 x 2.18 x 10-18 )
= 2 x 10-24 kg•m / sec
px / p = 10-24 / ( 2 x 10-24 )
= 0.5 or 50% uncertainty
classical concept of determinism failed in atomic phenomena
Bohr and Heisenberg
測不準的原因
( Physical Origin of the Uncertainty Principle )
Measuring process itself introduces the uncertainty–Heisenberg
electron
例 : to see an e- with a hypothetical microscope with one photon
objective lens
p p
-p sin -p sin
photon
scattered photon within 2will be detected
uncertainty in px of the photon :
px ( photon ) = 2 pphoton •sin
= 2 ( h/) sin
from momentum conservation :
px ( e- ) = 2 pphoton •sin= 2 ( h/) sin
introduced an uncertainty in e- momentum
to reduce px
increase ( e.g. use -wave, radio wave )
reduce
increase the uncertainty of e- position
( for high resolution small and large slit width )
Measuring process itself introduces the uncertainty
物質波與測不準原理 應改名為 “不準確原理”
( Matter Waves and the Uncertainty Principle ) ( –中文譯為“測不準原理”並不是很好的翻譯 )
•如果用波函數來描述粒子的運動行為 , 則一定無法同時準確的描述粒子的
位置及動量
•例如 : 一個在一維空間中以等速率運動的粒子 : ( x,t ) = A sin( kx –t )
(1) same amplitude A at all points
(2) well-defined = 2/k and = /2
(3) travels in “+x”direction , v = •= / k
this particle :
(1) has a well-defined p= h/動量的不確定性 , px = 0
(2) P( x,t ) dV = 2 dV = A2dV at any point ( Particle can be
found with equal probability at any point , 因為粒子以等速率
自–移動至 +處 , 因此在各點出現及停留的時間都一樣)
位置的不確定性 , x =
•但只有對週期性的波動而言 , 才有所謂的波長及頻率:
例如 : ( x,t ) = A sin( kx –t )在 t = 0 時 在 x = 0 那一點
( x,t=0 ) = A sin( kx ) ( x=0,t ) = A sin( -t )
( x,t=0 ) k = 2/
x
t
T = 1/
( x=0,t ) w = 2/ T
•三角弦波所描述的粒子都具有明確的動量及動能 :
p = h / , E = h或者反過來說 , 用來描述一個具有明確動量 ( p ) 及動能 ( E ) 的粒子的波函數應該是一個三角弦波 , 例如 :
( x,t ) = A sin( kx-t ) k = 2/ , = h/p= 2 , = E/h
•用來描述一個具有明確動量 ( p ) 及動能 ( E ) 的粒子的波函數應該是一
個三角弦波 , 例如 :
( x,t ) = A sin( kx-t ) , k = 2/ , = h/p
= 2 , = E/h
( x,t=0 )
x
v p ( x,t=0 )
x
( x,t=0 )
x
v = 0
vx( x,y,t ) = A sin( kxx –t )
kx = 2/x , x = h/px
•將問題推廣到二維空間上 : 考慮一個在 x 軸上等速度運動的粒子
( x,y,t )
x
y
這樣一個波要如何定義它的波長或頻率 ?
( x,t=0 )
x
•再來看另一種類型的波函數
( x,t=0 )
x
sin( kx ) , k = 2/
x
x
x
此波函數既不是單純的( 0 ) ,
也不是單純的( )
此類波函數是由一組不同波長的成分波組合而成的 :
x
•再來看另一種類型的波函數
( x,t=0 )
x此波函數既不是單純的( 0 ) ,
也不是單純的( )x
(1) more values of
less well-defined momentum, px ( px = h/)
(2) but narrower width of the resulting wave packet
better localized particle , x
(3) consistent with uncertainty principle( x,t=0 )2
xx
此類波函數是由一組不同波長的成分波組合而成的 :
•to describe a partially localized particle :
mix a large number of sinusoidal traveling waves
( x,t ) = A( k,) sin( kx –t )
( x,t ) = A( k,) sin( kx –t ) dk d
k0
0
例 : fix o
( x,t ) = A( k,o ) sin( kx –ot ) dk
= A( k ) sin( kx –ot ) dk
o
o
matter wave
particle packet
greater range of k
greater range of ( = 2/ k )
less well-defined momentum, px ( px = h/)
narrower width of the resulting wave packet
better localized particle , x
consistent with uncertainty principle
波包行進的速度 –群速
( Velocity of Wave Packet –Group Velocity )
group velocity velocity of the particle
例 : 1 = A sin( kx –t ) , 2 = A sin[ ( k+k ) x –( +) t ]
assume k << k , <<
( x,t ) = 1 + 2
= A sin[ ( k+k ) x –( +) t ] + A sin( kx –t )
sin a + sin b = 2 cos (a-b)/2 sin (a+b)/2
( x,t ) = 2A cos ( —————— ) sin [ ——————————— ]
= 2A cos ( —————— ) sin( kx –t )
k x –t2
( 2k+k ) x –( 2+t2
k x –t2
The envelop travels at vgroup
vgroup = ——— = / k d/ dk
= h / p
= 2/ k
= E / h
= / 2
vgroup = d/ dk = dE / dp = d( —— ) / d p
= 2p / 2m = m vparticle / m = vparticle
vgroup = vparticle
k = p / h dk = dp / h
= E / h d= dE / h
@ t=0
sin kx
cos kx/2x
= 2/k , 2/(k/2) >> 2/k
/2k/2
p2
2m
vgroup
vgroup
互補原則 ( Bohr’s Principle of Complementary )
Bohr’s Principle of Complementary :
The particle and the wave models are complementary
•No measurements can simultaneously reveal the particle and the
wave properties of matter.
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