１７ 不完全性定理 #宇宙 #不完全性YouTubeを始めました。【スタジオ・グーダ！】７日で若返りたい！#若返り #脳梗塞https://youtu.be/e5Vq60aEEnI
○正しいとも正しくないとも判定できない命題が存在する。・・・1930年にゲーデル（オーストリア 1906－78）が証明した。・・・その後、チューリングが、ある命題が真偽を判定できない命題であるかどうかを、あらかじめチェックする統一的な方法がないということを証明した。第1不完全性定理 自然数論を含む帰納的公理化可能な理論が、ω無矛盾であれば、証明も反証もできない命題が存在する（＝不完全）。＝一部、完全性と無矛盾性は両立できない命題がある。第2不完全性定理 自然数論を含む帰納的公理化可能な理論が、無矛盾であれば、自身の無矛盾性を証明できない。☆検証不能であり、かつ検証不能なことをあらかじめチェックできない命題があるとすれば、「探索は永遠である」ということになる。・・・探索を一生続けても何も残らないかもしれないが、・・・☆すべてのことを一つの理論で説明できないのなら、複数の理論を使えばいい。【参 照】1. 藤原正彦／小川洋子「世にも美しい数学入門」20050406 (p.147～151)2.不完全性定理 - Wikipedia3. ゲーデルの不完全性定理をお教えください。 - 知恵袋 - Yahoo! JAPANhttps://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1410933224
【更新履歴】20170429 定理の追加20170706 複数の理論を使うことの提案
１６ Uncertainty Principle #universe #uncertainty１６－１ 1927 Werner HeisenbergIntroduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. You can answer only according to probability on the level of quantum.○The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp σｘσｐ≧ｈ：No matter how improve the accuracy of the measurement, the product of the standard deviation of the position and momentum is greater than h.：One is uncertain when you try to confirm the other.・ｘ：position、ｐ：momentum・h：Planck constant（6.626×10^-34 J・s）: Particles are spread out within a certain range and exist on the level of quantum.○In the case of energy and time σＥσｔ≧ｈ：Beating only a little law of conservation of energy, it may be borrowed energy. ※However it must return as fast as borrow a lot.・Ｅ：energy、ｔ： time⇒Energy is possible to materialize.⇒Pairs of matter and anti-matter particles are constantly being created and annihilated.＝Quantum fluctuations ⇒Also in vacuum, pairs of matter and anti-matter particles are constantly being created and annihilated.: Vacuum fluctuations⇒The vacuum has a vastly complex structure.・Vacuum fluctuations causes a measurable force (Casimir effect) in infinitesimal.
※Vacuum fluctuations [3]１６－２ Casimir effect○The typical example is of the two uncharged conductive plates in a vacuum, placed a few nanometers apart. The Casimir effect produces the negative pressure. It is due to quantum vacuum fluctuations of the electromagnetic field.○As the two mirrors move closer to each other, the longer waves will no longer fit--the result being that the total amount of energy in the vacuum between the plates will be a bit less than the amount elsewhere in the vacuum. Thus, the mirrors will attract each other, just as two objects held together by a stretched spring will move together as the energy stored in the spring decreases. [4]○The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror. [5] Further, there may help development of quantum computers.
１６－３ Phase Transitions of Vacuum ／Inflation Cosmology／Quantum Tunneling○Vacuum is to shift to more of the state of low energy:phase transition.○In physical cosmology, cosmic inflation, cosmological inflation, or just inflation is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from 10 minus 36 square seconds until after 10 minus 34 square seconds.（The size of the universe (10^-27m) is much smaller than atom (10^-10m).⇒3 millimeters (10^-3m)）○The driving force behind this inflation has been said that the energy that was released during the phase transition of the vacuum.○In early inflationary models, the phase transition of the vacuum was considered that it is generated by the Quantum tunnelling or tunnel effect. However, in a model named new inflation or slow-roll inflation, instead of tunneling out of a false vacuum state, inflation is considered that occurred by rolling down a potential energy hill.※Quantum tunnelling or tunnel effect to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classicallycould not surmount. In the quantum theory it is explained by the uncertainty principle.It has important applications to modern devices such as the tunnel diode, quantum computing, and the scanning tunnelling microscope. [3]※α decay and tunnel effectCalculating the position of alpha particles by solving the Schlesinger equation, the α particles can also be present outside the atoms. Actually in α decay, α particles can go through the walls of energy at the edge of nuclei, digging tunnels.※Nuclear fusion is also thanks to the tunnel effect. Without the tunnel effect protons can not collide with each other. And the sun can not shine. [6]【Refereces】1. Uncertainty principle －Wikipedia 2. ハイゼンベルクの不確定性原理 - nifty Uncertainty principle- nifty3. RevoScience”Young researcher proposes new explaination for unsolved problems in physics…”4. “What is the Casimir effect?”SCUENTIFIC AMERICAN 19985. Casimir effect －Wikipedia6. Masahiro Maeno “Introduction to Quantum mechanics(Japanese) " 20060216 (p.107) 【Change log】20170930 Addition about tunnel effect
１６ 不確定性原理 #宇宙 #不確定性１６－１ １９２７年 ハイゼンベルク量子のレベルでは、ある粒子（たとえば電子）について、その位置と運動量は確率でしか答えられないという原理。○位置の標準偏差σxと運動量の標準偏差σpを結び付ける不等式 σｘσｐ≧ｈ：どんなに測定の精度を高めても、位置と運動量の標準偏差の積はｈよりも大きくなる。：片方を確定しようとすると片方が不確定になる。・ｘ：位置、ｐ：運動量・h：プランク定数（6.626×10^-34ジュール・秒）：粒子はある範囲の中に広がって存在している。○エネルギーと時間の場合 σＥσｔ≧ｈ：少しだけエネルギー保存則を破って、エネルギーを借りても良い。※ただしたくさん借りるほど早く返さなければならない。・Ｅ：借りるエネルギーの量、ｔ：借りる時間⇒エネルギーは物質化する。⇒粒子・反粒子が現れては消えている。[3]＝量子ゆらぎ⇒真空でも、粒子・反粒子が現れては消えている。：真空のゆらぎ⇒真空はただのゼロではなく、正と負のエネルギーがせめぎあい全体としてプラス・マイナスでゼロになっている。・真空のゆらぎは極微において計測可能な力（カシミール効果）を引き起こす。
図：真空のゆらぎ [3]１６－２ カシミール効果(Casimir effect) ○非常に小さい距離を隔てて設置された二枚の平面金属板が真空中で互いに引き合う現象を、静的カシミール効果という。○静的カシミール効果の引力作用は、二枚の金属板の間には真空エネルギーの一部の波しか入り込むことができないことから、真空エネルギーの密度が外側よりも減少することにより生じる。○また、二枚の金属板を振動させると光子やエネルギーが生じる。これを動的カシミール効果という。 [4] ○動的カシミール効果では、大量のエネルギーを要するが、理論的には電子や陽子などの他の粒子も、真空から生成することが可能。また、量子コンピュータの開発などに役立つ可能性がある。[5]
１６－３ 真空の相転移／インフレーション宇宙／トンネル効果 [6]○真空はよりエネルギーの低い状態の真空へと移行（相転移）する。○現在の宇宙論では、宇宙誕生の10^-36秒から10^-34秒後の間に、インフレーションと呼ばれる指数関数的な急激な膨張（原子(1e-10ｍ)よりはるかに小さい実宇宙（1e-27ｍ）⇒３ミリ程度（1e-3ｍ））があったとされている。○このインフレーションの原動力となったのは、真空の相転移の際に解放されたエネルギーだとされている。○初期のインフレーション理論では、偽の真空と真の真空の間に明確なポテンシャルの障壁があり、それをトンネル効果によって乗り越えることで真空の相転移が発生すると考えられていたが、新しいインフレーション理論（ゆっくり転がるインフレーション）では、明確なポテンシャルの障壁はなく、偽の真空から真の真空へと至る緩やかなポテンシャルの坂があるとされている。※トンネル効果：エネルギーの壁を、それより低いエネルギーを持った粒子が通り抜けてしまう現象。古典的に考えればトンネルを掘らない限りは不可能に思えるが、量子論では不確定性原理により、あたかもトンネルを掘ったかのように障壁を乗り越えてしまうことがある。半導体はこの原理を利用してつくられている。[7]※α崩壊とトンネル効果シュレジンガー方程式を解いてアルファ粒子の位置を計算すると、α粒子が原子の外側にも存在できる：原子核の縁のエネルギーの壁を、トンネルを掘るようにして通り抜けることができる。[8]※核融合もトンネル効果のおかげ。トンネル効果がなければ、陽子同士は衝突できない。太陽が輝くこともない。[9]【参考】1. 不確定性原理 - Wikipwdia 2. ハイゼンベルクの不確定性原理 - nifty3. 島根大学集中講義 真空の性質20094. カシミール効果 - Wikipedia 5.SJN news” チャルマース工科大、真空から光子を生成。「動的カシミール効果」を実験で確認”(2011)6. 偽の真空 – Wikipwdia7. トンネル効果 - Wikipwdia 8. 佐藤勝彦”「量子論」を楽しむ本” 9.前野昌弘 “量子力学入門" 20060216 (p.107) 【更新履歴】20170930 トンネル効果について追記
１５ Quantum Mechanics #universe #mechanics１５－１ What is Quantum mechanics？○Word root：Energy, length, speed, and time, those physical quantities have the smallest unit(quanta) that can not be divided any more. ○About the particles like the size of electron (: 2.8 × 10 ^ -15m), you can see the state (the value of the energy), if you get the wave function by solving the wave equation (Schrodinger equation: partial differential equation 1926) , or if you solve Heisenberg`s equation (the determinant 1925) .：By differentiating the wave function, we know the values of energy and momentum.※The wave equation and Heisenberg`s equation is mathematically equivalent. Since Heisenberg`s equation is more complicated than the wave equation, computation of concrete cases often uses wave equations. On the other hand, when dealing with the general theory of quantum mechanics Heisenberg`s equation is more suitable.○The Wave equation• The Wave equation is a Fundamental equation of quantum mechanics. It is contrasted with Newton's equation of motion of classical mechanics.・We can calculate the state of the matter wave stochastically by the equation.・The Wave equation cannot be obtained strictly by using classical mechanics or mathematics. It is the equation that successfully describes the physical phenomena.※Relativity is classical mechanics.
： reduced Planck constantｔ：timeｍ： mass of particleΨ（Psi）：wave function： Complex quantity, which was introduced to represent the waveX：position V： potential energy○The squared modulus of the wave function is a real number interpreted as the probability density of measuring a particle's being detected at a given place. ・The figure below shows the graph of the probability density and the energy state of matter. [1]
※The red line is the energy state(n). The probability density and n is changed at intervals.☆The universe had particles that have exactly the same properties without limit, at energy following the mathematical rules!○The universe was born from a quantum fluctuation.○Wave function is a function of the complex because the wave equation contains complex number. [2]・・・The universe is represented by imaginary!１５－２ Material Wave（de Broglie wave）○On a submicroscopic level, there is no boundary strict between particles and wave.→When existence is regarded as a wave motion, in physics, it is called a "place." For example, light is wave motion of an electric field and magnetic field.※ It is under a very microscopic situation like an electron beam that the character as a wave is actually observed.※ It is impossible to determine the size of the electron.※ And the length of orbit around the nucleus is an integer multiple of the wavelength (10 minus 10 square m).＝Why the electrons are not getting stuck in the nucleus.○The image of the wave of complex number [1]
※Whole wave represents one of the particles.※Height and depth of the wave corresponds to the size of the wave function.：de Broglie wave＝the wave function○Material Wave：A wave that moves the space represented by the complex number. [3]：without direction（Scalar wave）
○In general, the substance obviously shows the property of waves is the world of elemental particles such as electrons composing atoms, protons and neutrons, the world of less than 10 millionths of a millimeter (10 ^ -10 m).This is one tenth of a world of nanometer (one millionth of a millimeter (10 ^ - 9 m)) called nanotechnology etc.And when becoming a world equal to or larger than an atom, the property as a wave hardly appears.• However, in the world beyond atoms, the nature of waves may appear.・One example is the superfluid phenomenon of helium at cryogenic temperature. This is called "Bose-Einstein condensation", and the waves of a plurality of atoms overlap each other at cryogenic temperature, thereby exhibiting properties like Bose particles, and the viscosity becomes zero.※Superfluid refers to a state in which the viscous resistance of a liquid disappears at a constant flow rate or less. There are strange phenomena, such as the liquid climbing the wall by the attraction of the container atom and flowing out from the narrow gap which can not be passed by ordinary liquid. ※Fermi particles (electrons, protons, neutrons) follow Pauli exclusion principle , but Bose particles (photons) do not follow.・In 1999, the group of Vienna University succeeded to let C60 molecules with 60 carbon atoms gather in the form of soccer balls to draw interference fringes. １５－３ Applications [4]・Description of the behaviors of the subatomic particles・Computational chemistry・The presence description of antimatter and negative energy・Basic theory of semiconductor engineering → Electronic devices design（the laser, the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging (MRI) , the light emitting diode, USB drives）・Basic theory of superconductivity and superfluidity・Quantum cryptography・Quantum computing・Quantum teleportation・Parallel universes○Unsolved problem・Quantum gravity theory is incomplete.１５－４ Uncertainty Principle → Ｎｏ．１６【References】1. 広江克彦”EMANの量子力学” Katsuhiko Hioe” EMAN’s Quantum mechanics”(Japanese)2.Lecture 3: The Wave Function - MIT OpenCourseWare3. EMAMの物理学・量子力学／調和振動子 EMAN’s Physics ・Quantum mechanics／ harmonic oscillator (Japanese)4. Quantum mechanics - Wikipedia【Change log】20170929 Addition about Material Wave, the wave equation and Heisenberg`s equation
