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De Broglie Wavelength Calculator

Calculate the de Broglie wavelength, momentum, and wave properties of particles

Parameters

kgⓘ
m/sⓘ
eVⓘ
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Animation

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Calculated Values

De Broglie Wavelength:
727.34;pm727.34;pm
Momentum:
9.1100e−25;kg⋅m/s9.1100e-25;kg·m/s
Kinetic Energy:
4.5550e−19;J4.5550e-19;J
Wave Number:
8638577143.41;m−18638577143.41;m⁻¹
Frequency:
687436126827000.13;Hz687436126827000.13;Hz

Visualization

De Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that associates a wavelength with any particle, given by λ = h/p, where h is Planck's constant and p is the particle's momentum. This concept was proposed by Louis de Broglie in 1924 and provided crucial insight into wave-particle duality.

The de Broglie wavelength represents the characteristic length scale over which quantum effects become important for a particle. For macroscopic objects, this wavelength is extremely small, making quantum effects negligible. However, for microscopic particles like electrons, the wavelength can be comparable to atomic dimensions.

The wavelength can be expressed in terms of velocity as λ = h/(mv), where m is the particle's mass and v is its velocity. For relativistic particles, the relativistic momentum p = γmv must be used, where γ = 1/√(1 - v²/c²) is the Lorentz factor.

The de Broglie wavelength is crucial for understanding phenomena like electron diffraction, quantum tunneling, and the behavior of particles in quantum wells and quantum dots. It's fundamental to technologies like electron microscopes, quantum computing, and nanotechnology.

When the de Broglie wavelength is comparable to the size of obstacles or apertures, particles exhibit wave-like behavior such as interference and diffraction. This is the basis for many quantum mechanical experiments and applications.

Key Concepts

  • De Broglie Wavelength: λ = h/p = h/(mv)
  • Momentum: p = mv (non-relativistic)
  • Relativistic Momentum: p = γmv
  • Wave-Particle Duality: Particles exhibit both particle and wave properties
  • Quantum Effects: Important when λ ≈ size of system
  • Interference: Particles can interfere with themselves

Real-World Applications

  • Electron Microscopy: Imaging at atomic resolution
  • Quantum Computing: Qubit manipulation and control
  • Nanotechnology: Quantum confinement effects
  • Particle Physics: Understanding fundamental particles
  • Materials Science: Electron diffraction analysis

Physics Equations

De Broglie Wavelength:
λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}
Momentum:
p=mvp = mv
Kinetic Energy:
Ek=12mv2E_k = \frac{1}{2}mv^2
Wavelength from Energy:
λ=h2mEk\lambda = \frac{h}{\sqrt{2mE_k}}
Wave Number:
k=2πλk = \frac{2\pi}{\lambda}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Momentum

First, we calculate the particle's momentum:

Equation:

p=mvp = mv

Calculation:

p=9.11e−31×1.00e+6=9.11e−25 kg\cdotpm/sp = 9.11e-31 \times 1.00e+6 = 9.11e-25 \text{ kg·m/s}

Explanation:

Momentum is the product of mass and velocity.

2

Step 2: Calculate De Broglie Wavelength

The de Broglie wavelength is related to momentum by:

Equation:

λ=hp\lambda = \frac{h}{p}

Calculation:

λ=6.63e−349.11e−25=727.34 pm\lambda = \frac{6.63e-34}{9.11e-25} = 727.34 \text{ pm}

Explanation:

This gives the characteristic wavelength associated with the particle.

3

Step 3: Calculate Kinetic Energy

The kinetic energy of the particle is:

Equation:

Ek=12mv2E_k = \frac{1}{2}mv^2

Calculation:

Ek=12×9.11e−31×(1.00e+6)2=4.56e−19 JE_k = \frac{1}{2} \times 9.11e-31 \times (1.00e+6)^2 = 4.56e-19 \text{ J}

Explanation:

This is the energy due to the particle's motion.

4

Step 4: Calculate Wavelength from Energy

Alternatively, we can calculate wavelength from energy:

Equation:

λ=h2mEk\lambda = \frac{h}{\sqrt{2mE_k}}

Calculation:

λ=6.63e−342×9.11e−31×1.60e−19=1226.38 pm\lambda = \frac{6.63e-34}{\sqrt{2 \times 9.11e-31 \times 1.60e-19}} = 1226.38 \text{ pm}

Explanation:

This method uses the particle's kinetic energy instead of velocity.

Frequently Asked Questions

What is the de Broglie wavelength?

The de Broglie wavelength is the wavelength associated with any particle, given by λ = h/p, where h is Planck's constant and p is the particle's momentum. It represents the wave-like nature of particles.

Why don't we see quantum effects for macroscopic objects?

For macroscopic objects, the de Broglie wavelength is extremely small (much smaller than atomic dimensions), making quantum effects negligible. The wavelength scales as 1/mass, so larger objects have smaller wavelengths.

How does the de Broglie wavelength relate to energy?

For non-relativistic particles, λ = h/√(2mE_k), where E_k is the kinetic energy. Higher energy particles have shorter wavelengths, making quantum effects more pronounced.

What is wave-particle duality?

Wave-particle duality is the concept that particles can exhibit both particle-like properties (localized position, momentum) and wave-like properties (interference, diffraction). The de Broglie wavelength quantifies the wave aspect.

When do quantum effects become important?

Quantum effects become important when the de Broglie wavelength is comparable to the size of the system or obstacles. This typically occurs for microscopic particles or in confined systems.

Practice MCQs

  1. The de Broglie wavelength of a particle is:
  2. If the momentum of a particle is doubled, its de Broglie wavelength becomes:
  3. For a particle with mass m and velocity v, the de Broglie wavelength is:
  4. Which particle would have the longest de Broglie wavelength?
  5. The de Broglie wavelength is important for understanding: