Change position uncertainty Δx to see how momentum uncertainty Δp responds — visualized as a Gaussian wave packet in real time. Intuitively grasp the fundamental principle of quantum mechanics.
Particle Presets
Wave Packet Parameters
位置 不確定性 Δx
pm
中心位置 x₀ (相対)
pm
中心運動量 p₀ (log₁₀)0
ΔxΔp / (ℏ/2)
1.000
（≥ 1.000 が原理 要求）
Position uncertainty Δx
Momentum uncertainty Δp (minimum)
Δx
— pm
Δp (最小)
— eV/c
最小運動Energy
— eV
ΔE (線幅)
— eV
Wave
Mom
Tradeoff
Theory & Key Formulas
位置-運動量：$\Delta x \cdot \Delta p \geq \dfrac{\hbar}{2}$
A fundamental principle of quantum mechanics: $\Delta x \cdot \Delta p \geq \hbar/2$ ($\hbar = h/(2\pi) \approx 1.055 \times 10^{-34}$ J·s) always holds. The product of the position uncertainty Δx and momentum uncertainty Δp is at least ℏ/2. The equality holds only for a Gaussian wave packet (coherent state), which is called a minimum uncertainty state.
How is the uncertainty principle different from the measurement problem?
This is different from the measurement problem ("measurement disturbs the system"). The uncertainty principle is a property of the quantum state itself. Fourier transforming a position eigenstate (delta function) yields a state containing all momentum components, and this holds even before any measurement. Robertson's generalized theorem (1929) states that for any two non-commuting operators  and B̂, ΔA·ΔB ≥ |⟨[Â,B̂]⟩|/2.
What is the quantum confinement effect?
A phenomenon where confining a particle in a small space increases its kinetic energy due to the uncertainty principle. In quantum dots (semiconductor crystals a few to tens of nm in size), the band gap changes with confinement size, altering the emission wavelength. This is why "quantum dot color changes with size." This effect is also important in the design of miniaturized semiconductor devices (transistors below 10 nm).
How does an STM (scanning tunneling microscope) see atoms?
It measures the tunneling current of electrons across a vacuum gap (~1 nm) between the tip and the sample. Due to the uncertainty principle, electrons can pass through an energy barrier that is classically insurmountable (quantum tunneling). This tunneling current depends exponentially on the gap distance, allowing surface topography to be measured with ~0.01 nm precision, enabling the identification of individual atoms.
What is the relationship between DFT (first-principles calculation) and the uncertainty principle?
DFT (density functional theory) is a method that solves the electronic wave function to predict material properties. The kinetic energy term in the Kohn-Sham equations inherently incorporates the uncertainty principle. Many material constants used in CAE (elastic modulus, thermal expansion coefficient, magnetism) are obtained from DFT calculations, making it essential for designing and optimizing advanced materials (graphene, 2D materials, topological insulators) where quantum effects at the nanoscale are critical.
What is Heisenberg Uncertainty Principle?
Heisenberg Uncertainty Principle 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 Heisenberg Uncertainty Principle Visualizer. 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 Heisenberg Uncertainty Principle Visualizer 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.