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For those who are not shocked when they first come across quantum theory cannot possibly have understood it.

Niels Bohr, in 1952, quoted in Heisenberg, Werner (1971). Physics and Beyond. New York: Harper and Row. pp. 206.

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物質波的疊加態，以及糾纏態

與EPR悖論

國立屏東大學｜應用物理系劉岱泯

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物質波

Matter Wave

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物質 Matter

Credits: NASA

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波 Wave

https://www.allnews.tw/news/23232

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波的文學 Literature of Wave

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波濤洶湧

波瀾壯闊

一波三折

暗送秋波

軒然大波

波光粼粼

有關波的成語：

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光，是波還是粒子？！

Particle vs Wave

1670 ………….. 1801

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波粒二象性 Wave-particle duality

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Blind men and elephant

https://docs.google.com/presentation/d/1UoRaE61mLk4glDwXU5i42mAxVVSPXSnEXAUraMfIRhk/edit#slide=id.g100ed4cc898_0_20

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粒子：動量 Momentum

動量：質量×速度 Momentum = mv

https://docs.google.com/presentation/d/1UoRaE61mLk4glDwXU5i42mAxVVSPXSnEXAUraMfIRhk/edit#slide=id.g100ed4cc898_0_20

https://docs.google.com/presentation/d/1UoRaE61mLk4glDwXU5i42mAxVVSPXSnEXAUraMfIRhk/edit#slide=id.g100ed4cc898_0_20

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光子的動量 Photon momentum

Momentum = mv

光子質量 ＝ 0 (?)

光子動量 ＝ 0xc = 0

!!! ???

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光電效應 Einstein’s theory of photon electric effect

momentum = mv

光子質量＝0

光子動量 ＝ 0xC = 0

https://docs.google.com/presentation/d/1UoRaE61mLk4glDwXU5i42mAxVVSPXSnEXAUraMfIRhk/edit#slide=id.gfa74b50b62_0_60

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光子的動量 Photon momentum

E = mc2

E＝hf

P ＝ mc

P= mc

∵ m＝ E/c2

P= E/c2×c= E/c

∵ E= hf

P= hf/c= h ÷ (c/f) ∵ c= ℷf

= h/ℷ

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波粒二象性 Wave-particle duality

P= h/ℷ

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光具有波粒二象性 Wave-particle duality

波粒二象性只有光子獨享？

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Louis Victor Pierre Raymond, 7th Duc de Broglie

In his 1924 PhD thesis, he postulated the wave nature of electrons and suggested that all matter has wave properties.

https://en.wikipedia.org/wiki/Louis_de_Broglie

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換邊發球

ℷ= h/P

= h/mv

https://www.womenshealthmag.com/tw/fitness/celeb-workoutsecret/g37126000/tokyo2020-10-things-to-know-tai-tzuying/

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物質波波長

ℷ= 6.63x10-34/6.63= 1.00x10-34 m

換邊發球

王建民在球員生涯中的最快球速為2004年於小聯盟比賽中投出的99mph（約159km/h）

v= 159x1000/60/60= 44.2 m/s

m= 150g= 0.150 Kg

P= mv= 6.63 Kg∙m/s

ℷ= h/P

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電子的物質波波長

ℷ=6.63x10-34/(9.1 x10-34x2.9x108)= 2.5x10-9 m

v= 2.9 x108 m/s

m= 9.1 x10-34 Kg

ℷ= h/P

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電子的物質波

https://www.protoxrd.com/products/laue-single-crystal/laue-cos

https://chem.libretexts.org/Courses/Franklin_and_Marshall_College/Introduction_to_Materials_Characterization__CHM_412_Collaborative_Text/Diffraction_Techniques/Electron_Diffraction

x-ray

electron

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光學顯微鏡 vs 電子顯微鏡

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原子中的電子

Bohr Model

Schrödinger Model

https://commons.wikimedia.org

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波函數

三角函數 Sin(x)/Cos(x)

波函數 𝚿(x)

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https://www.youtube.com/watch?v=R5v8AIcx7Uo

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物質波的疊加態

Quantum Superposition

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波函數是解 Solutions of Shrödinger equation

態 State

方程式 Eauation

解 Solutions

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波動方程

v = 𝜆f

Sin(x,t)/Cos(x,t)

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Schrödinger Eq.

𝚿(x,t)

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線性系統 Linear system

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線性系統的解 Shrödinger equation and solutions

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⏐𝚿1⟩

⏐𝚿2⟩

⏐𝚿⟩ =𝛼⏐𝚿1⟩ + 𝛽⏐𝚿2⟩

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物質波的疊加態 Superposition of states

⏐𝚿⟩ =𝛼⏐𝚿1⟩ + 𝛽⏐𝚿2⟩ + ...

bra–ket notation, or Dirac notation

⏐𝚿⟩: bra

⟨𝚿⏐:ket

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Shrödinger’s cat

×

cat may be considered simultaneously

both alive and dead

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物質波的糾纏態

Quantum Entanglement

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糾纏 Entanglement

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糾纏的物理 Physical Entanglements

Engineering biosynthetic cell encapsulation systems

J.J. Roberts, P.J. Martens, in Biosynthetic Polymers for Medical Applications, 2016

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粒子的量子態 Quantum states of particles

2

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⏐u1⟩, ⏐u2⟩

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⏐d1⟩, ⏐d2⟩

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⏐u1⟩⊗⏐d2⟩, ⏐u2⟩⊗⏐d2⟩..

1

2

1

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量子態的疊加 Superpositions of quantum states

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2

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𝛼1⏐u1⟩ + 𝛼2⏐u2⟩

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𝛽1⏐d1⟩ + 𝛽2⏐d2⟩

1

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兩個粒子的量子態 Quantum states of particles

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(𝛼1⏐u1⟩ + 𝛼2⏐u2⟩) ⊗ (𝛽1⏐d1⟩ + 𝛽2⏐d2⟩)

2

1

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兩個粒子的量子態 Quantum states of particles

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𝛼1𝛽1⏐u1⟩⊗⏐d1⟩ + 𝛼1𝛽2⏐u1⟩⊗⏐d2⟩

+

𝛼2𝛽1⏐u2⟩⊗⏐d1⟩ + 𝛼2𝛽2⏐u2⟩⊗⏐d2⟩

2

1

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兩個粒子的疊加態 Quantum superpostion of particles

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(⏐u1⟩⊗⏐d1⟩) ＋ (⏐u2⟩⊗⏐d2⟩)

?

=

(...........)1⊗(...........)2

2

1

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兩個粒子的疊加態 Quantum superpostion of particles

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(1⏐u1⟩⊗⏐d1⟩) ＋ (1⏐u2⟩⊗⏐d2⟩)

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𝛼1𝛽1⏐u1⟩⊗⏐d1⟩ + 𝛼1𝛽2⏐u1⟩⊗⏐d2⟩

+

𝛼2𝛽1⏐u2⟩⊗⏐d1⟩ + 𝛼2𝛽2⏐u2⟩⊗⏐d2⟩

2

1

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無解 No solutions!!

𝛼1𝛽1 = 1

𝛼2𝛽2 = 1

𝛼1𝛽2 = 0

𝛼2𝛽1 = 0

(⏐u1⟩⊗⏐d1⟩) ＋ (⏐u2⟩⊗⏐d2⟩)

?

=

(...........)1⊗(...........)2

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https://www.youtube.com/watch?v=G3HSP3qMgKI

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飛鴿傳書 Navigation with Quantum Entanglement

https://www.youtube.com/watch?v=0A1ouV7iD8o

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EPR 悖論

EPR paradox

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Einstein–Podolsky–Rosen paradox

http://www.drchinese.com/David/EPR.pdf

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Classical entaglement

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Quantum entaglement

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EPR paradox: Start from 1'52"

https://www.youtube.com/watch?v=5HJK5tQIT4A

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總結 Summary

  • 粒子具有波粒二象性。
  • 物質波在量子力學裡用波函數來表示。
  • 波函數是Schrodinger Eq.的解，解的波函數疊加後也是解。
  • 量子糾纏性是量子力學中獨特的特性。

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Einstein was confused, not the quantum theory.

Stephen Hawking,

Lecture at the Amsterdam Symposium on Gravity, Black Holes, and String Theory (June 21, 1997)