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量子力学

Wave-Particle Duality Simulator

Calculate de Broglie wavelength from particle mass and velocity and visualize double-slit interference patterns. Also compute electron microscope resolution. Experience the quantum strangeness of matter waves numerically.

Particle Presets

Particle Parameters

質量 m (log₁₀ kg) —
Velocity v (m/s, log₁₀) —

Double-Slit Setup

Slit間隔 d (nm)
nm
Screen距離 L (mm)
mm
de Broglie Wavelength λ
—
m
運動量 p
—
運動Energy
—
縞間隔 Δy
—
v/c 比
—
Interference
Theory & Key Formulas
$\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$

$h = 6.626 \times 10^{-34}$ J·s
縞間隔：$\Delta y = \dfrac{\lambda L}{d}$

光子波長：$\lambda = hc/E$, $c = 3 \times 10^8$ m/s

🎓 Learn Wave-Particle Duality Through Dialogue

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電子が「波でもWith粒子でもある」って言われても、なんかPinとこないんよね。波だったらSpreadがあるはずで、粒子だったら点 はずで…どっちなんか？
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古典的な「波」か「粒子」かどちらかに分類しようとする が間違いで、電子は「電子」という量子力学的な存在なんだ。伝播 ときは波動関数が広がって（干渉する）、観測するときは1点で検出される（粒子的）。二重Slit実験を1個ずつ電子を撃っても、多数集めると干渉縞ができる——これが現実に観測されている事実だよ。
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ド・ブロイ波長 λ=h/mv って、Velocityが速いほど波長が短くなるんね。じゃあ電子を加速すれば波長を短くできる？電子顕微鏡はそれを使ってるんか？
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完全にそう！透過型電子顕微鏡（TEM）は電子を100〜300kVで加速する。100kV 電子 波長は約3.7pm、可視光（500nm） 10万分 1以下だ。分解能 目安はλ/2程度な で、原子間距離（0.1〜0.3nm）を余裕で分解できる。Sliderーで「電子 100keV」を選んで確認してみよう。
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じゃあ野球Ballにもド・ブロイ波長があるんか？でも野球Ballが干渉するなんて聞いたことないんが。
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理論的にはある。でも野球Ball（145g、30m/s） λは約1.5×10⁻³⁴m。これはPlanck長（10⁻³⁵m）と同程度で、陽子 10²⁰分 1以下 大きさ。こんなScaleで干渉を生む「Slit」は物理的に作れないし、環境と 相互作用による量子デコヒーレンスで瞬時に波動性が消える。これがマクロな物体に量子効果が現れない理由だ。
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CAEや材料工学と 関係はWithか？量子力学って実務で出てきか？
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材料Simulationで超重要だよ。第一原理計算（DFT：密度汎関数理論）は電子 波動関数を解いて材料 物性（格子定数、弾性定数、電気抵抗、磁性）を予測する。CAE 材料DataBase 多くはDFT計算from 来ている。半導体デバイス Simulation（MOSFET 量子tonネルCurrentなど）も波動性が本質的な役割を果たす分野だ。

Frequently Asked Questions

How was it confirmed that electrons have wave-like properties?
In 1927, Davisson and Germer fired electrons at a nickel crystal and observed a diffraction pattern, confirming de Broglie's prediction. Thomson independently verified it the same year. Today, even when electrons are fired one by one in a double-slit experiment, an interference pattern emerges after thousands accumulate, directly proving the wave nature of a single electron.
In the double-slit experiment with electrons, why does the interference pattern disappear when we observe which slit the electron goes through?
Observation always involves an interaction between the electron and something else (a photon, an electric field, etc.). This interaction entangles the electron's wavefunction, breaking the superposition state (the state of passing through both slits). This is called quantum decoherence. As soon as information about 'which path' leaks into the environment, interference disappears.
Why can an electron microscope see atoms?
The resolution limit depends on the wavelength of the probe (Abbe diffraction limit). Visible light (400–700 nm) can never resolve atoms (diameter ≈ 0.1–0.5 nm). A transmission electron microscope (TEM) accelerates electrons to 200–300 kV, reducing the de Broglie wavelength to a few picometers, easily resolving interatomic distances (0.1–0.3 nm). With a spherical aberration corrector (Cs corrector), resolution drops below 50 pm.
Do relativistic effects affect the de Broglie wavelength?
When electrons are accelerated to high energies (tens of keV or more), their speed approaches the speed of light, requiring relativistic corrections. The relativistic momentum is $p = \gamma m_0 v$ (where $\gamma$ is the Lorentz factor), resulting in a shorter wavelength than non-relativistic calculations. At 200 kV in a TEM, electrons reach about 70% of the speed of light, and the relativistic correction introduces a difference of about 3%.
How are first-principles calculations (DFT) related to the de Broglie wavelength?
DFT (density functional theory) self-consistently solves for the electron wavefunctions (Kohn-Sham orbitals) in a material. The de Broglie wavelength of electrons is often comparable to the material's lattice constant (0.1–0.5 nm), and resonance/scattering with the lattice determines electrical resistivity, thermal conductivity, and magnetism. A significant portion of the structural material data (elastic constants, thermal expansion coefficients) used in CAE is supplied by DFT calculations.

What is Wave-Particle Duality?

Wave-Particle Duality is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Physical Model & Key Equations

The simulator is based on the governing equations behind Wave-Particle Duality Simulator-. Understanding these equations is key to interpreting the results correctly.

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Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.

Real-World Applications

Engineering Design: The concepts behind Wave-Particle Duality Simulator- are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.

Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.

CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.

Common Misconceptions and Points of Caution

Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.

Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.

Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.