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.