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Wave-Particle Duality Calculator

Calculate wave-particle duality properties including de Broglie wavelength, photon energy, momentum, and frequency

Parameters

kgⓘ
m/sⓘ
nmⓘ
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Calculated Values

De Broglie Wavelength:
7.2734e−10;m7.2734e-10;m
Particle Momentum:
9.1100e−25;kg⋅m/s9.1100e-25;kg·m/s
Particle Kinetic Energy:
4.5550e−19;J4.5550e-19;J
Photon Energy:
3.9729e−19;J3.9729e-19;J
Photon Momentum:
1.3252e−27;kg⋅m/s1.3252e-27;kg·m/s
Photon Frequency:
599584915999999.88;Hz599584915999999.88;Hz

Visualization

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics where all particles exhibit both wave-like and particle-like properties. This dual nature is essential for understanding quantum phenomena.

The de Broglie wavelength λ = h/p relates the wavelength of a particle to its momentum, where h is Planck's constant and p is the particle's momentum. This shows that matter has wave-like properties.

For photons, the energy is related to frequency by E = hν = hc/λ, where ν is frequency, λ is wavelength, and c is the speed of light. Photons also have momentum p = h/λ = E/c.

The uncertainty principle ΔxΔp ≥ ℏ/2 shows that we cannot simultaneously know both the position and momentum of a particle with arbitrary precision. This is a consequence of wave-particle duality.

Wave-particle duality has been demonstrated in experiments like the double-slit experiment, where particles show interference patterns characteristic of waves while maintaining particle-like detection.

Key Concepts

  • De Broglie Wavelength: λ = h/p = h/mv
  • Photon Energy: E = hν = hc/λ
  • Photon Momentum: p = h/λ = E/c
  • Uncertainty Principle: ΔxΔp ≥ ℏ/2
  • Wave Function: ψ(x,t) describes particle behavior
  • Probability Density: |ψ|² gives detection probability

Real-World Applications

  • Electron Microscopy: Matter wave imaging
  • Quantum Computing: Wave function manipulation
  • Spectroscopy: Photon energy analysis
  • Particle Accelerators: Matter wave interference
  • Quantum Optics: Photon statistics

Physics Equations

De Broglie Wavelength:
λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}
Photon Energy:
E=hν=hcλE = h\nu = \frac{hc}{\lambda}
Photon Momentum:
p=hλ=Ecp = \frac{h}{\lambda} = \frac{E}{c}
Particle Momentum:
p=mvp = mv
Uncertainty Principle:
ΔxΔp≥ℏ2\Delta x\Delta p \geq \frac{\hbar}{2}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate De Broglie Wavelength

The wavelength associated with the particle:

Equation:

λ=hmv\lambda = \frac{h}{mv}

Calculation:

λ=6.626e−349.110e−31×1.000e+6=7.273e−10 m\lambda = \frac{6.626e-34}{9.110e-31 \times 1.000e+6} = 7.273e-10 \text{ m}

Explanation:

This gives the wavelength associated with the particle's wave-like behavior.

2

Step 2: Calculate Particle Momentum

The classical momentum of the particle:

Equation:

p=mvp = mv

Calculation:

p=9.110e−31×1.000e+6=9.110e−25 kg\cdotpm/sp = 9.110e-31 \times 1.000e+6 = 9.110e-25 \text{ kg·m/s}

Explanation:

This gives the particle's momentum in the classical sense.

3

Step 3: Calculate Photon Energy

The energy of the photon:

Equation:

E=hcλE = \frac{hc}{\lambda}

Calculation:

E=6.626e−34×2.998e+85.000e−7=3.973e−19 JE = \frac{6.626e-34 \times 2.998e+8}{5.000e-7} = 3.973e-19 \text{ J}

Explanation:

This gives the energy carried by the photon.

4

Step 4: Calculate Photon Momentum

The momentum of the photon:

Equation:

p=hλp = \frac{h}{\lambda}

Calculation:

p=6.626e−345.000e−7=1.325e−27 kg\cdotpm/sp = \frac{6.626e-34}{5.000e-7} = 1.325e-27 \text{ kg·m/s}

Explanation:

This gives the momentum carried by the photon.

Frequently Asked Questions

What is wave-particle duality?

Wave-particle duality is the fundamental property of quantum objects to exhibit both wave-like and particle-like behavior depending on how they are observed or measured.

What is the de Broglie wavelength?

The de Broglie wavelength λ = h/p relates the wavelength of a particle to its momentum, showing that matter has wave-like properties.

How do photons differ from matter particles?

Photons are massless particles that always travel at the speed of light. Their energy is E = hν and momentum is p = h/λ, while matter particles have mass and can travel at any speed below c.

What is the uncertainty principle?

The uncertainty principle ΔxΔp ≥ ℏ/2 states that we cannot simultaneously know both the position and momentum of a particle with arbitrary precision.

Why is wave-particle duality important?

It's fundamental to quantum mechanics and explains phenomena like interference patterns, quantum tunneling, and the behavior of particles at the atomic scale.

Practice MCQs

  1. The de Broglie wavelength of a particle is:
  2. For a photon, the energy is related to wavelength by:
  3. Which particle has the longer de Broglie wavelength at the same velocity?
  4. The uncertainty principle relates:
  5. What happens to the de Broglie wavelength as velocity increases?