4. Quantum Tunneling

Calculating the Wavelength of Particles and Photons ⚛️🌊🔬

Particles and photons exhibit wave properties, and their wavelengths are linked to energy and momentum. Let’s explore the formulas for matter and electromagnetic radiation, and perform some calculations! 🚀📡

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1. Wavelength Formulas for Matter and Radiation

(A) de Broglie Wavelength (Matter Waves) 🌊⚛️

For a moving particle, the de Broglie wavelength is:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where:

• $\lambda$ = de Broglie wavelength (m)
• $h = 6.63 \times 10^{-34}$ J·s (Planck’s constant)
• $p = mv$ = momentum (kg·m/s)
• $m$ = mass of the particle (kg)
• $v$ = velocity of the particle (m/s)

✔ Used for electrons, protons, and atoms in quantum mechanics.

📌 Example: Electrons in atoms have de Broglie wavelengths comparable to atomic radii.

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(B) Photon Wavelength: Energy and Momentum Relationship 📡

Photons (light particles) have no mass but still carry momentum and energy.

✔ Energy of a photon:

$E = h f = \frac{hc}{\lambda}$

✔ Momentum of a photon:

$p = \frac{E}{c} = \frac{h}{\lambda}$

​

where:

• $E$ = photon energy (J)
• $f$ = frequency (Hz)
• $c = 3.00 \times 10^8$ m/s (speed of light)
• $p$ = momentum (kg·m/s)

📌 Example: X-rays and gamma rays have shorter wavelengths and higher energy than visible light.

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2. Worked Examples: Particle and Photon Wavelengths

📌 Example 1: de Broglie Wavelength of an Electron

Find the wavelength of an electron moving at

$5.0 \times 10^6$

 m/s.
(Electron mass:

$9.11 \times 10^{-31}$

 kg)

✅ Solution:
Using:

$\lambda = \frac{h}{mv}$

$\lambda = \frac{6.63 \times 10^{-34}}{(9.11 \times 10^{-31}) (5.0 \times 10^6)}$

$\lambda = 1.45 \times 10^{-10} \text{ m} = 0.145 \text{ nm}$

🔹 Answer: The electron’s wavelength is 0.145 nm, similar to X-ray wavelengths.

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📌 Example 2: Wavelength of a Photon

Find the wavelength of a photon with an energy of 3.0 eV.

✅ Solution:
Convert energy to joules:

$1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$

$E = (3.0)(1.6 \times 10^{-19}) = 4.8 \times 10^{-19} \text{ J}$

Using:

$\lambda = \frac{hc}{E}$

$\lambda = \frac{(6.63 \times 10^{-34})(3.00 \times 10^8)}{4.8 \times 10^{-19}}$

$\lambda = 4.14 \times 10^{-7} \text{ m} = 414 \text{ nm}$

🔹 Answer: The photon’s wavelength is 414 nm, in the blue-violet range of visible light.

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📌 Example 3: Wavelength of a Baseball

A baseball (0.145 kg) moving at 40 m/s. Find its de Broglie wavelength.

✅ Solution:
Using:

$\lambda = \frac{h}{mv}$

$\lambda = \frac{6.63 \times 10^{-34}}{(0.145)(40)}$

​

$\lambda = 1.14 \times 10^{-34} \text{ m}$

🔹 Answer: The wavelength is

$1.14 \times 10^{-34}$

 m, too small to observe quantum effects.

📌 Conclusion: Large objects (e.g., baseballs) have negligible wave properties!

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3. Applications of Matter and Photon Wavelengths 🌍

✔ Electron Microscopes 🔬 – Use electron wavelengths for high-resolution imaging.
✔ X-ray & Gamma Ray Spectroscopy 📡 – Identifies atomic structures.
✔ Quantum Computing 💻 – Uses wave-like behavior of particles.
✔ Solar Panels ☀️ – Convert photon energy into electricity.

📌 Key Fact: Quantum mechanics dominates at atomic scales, where de Broglie wavelengths are significant!

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4. Key Takeaways! 🎯

✔ Particles have wave-like properties, described by de Broglie’s equation.
✔ Photons have energy and momentum, related to their wavelength.
✔ Electrons have wavelengths similar to atomic scales, enabling quantum effects.
✔ Macroscopic objects have negligible wavelengths, making quantum effects undetectable.

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5. Want to Try a Challenge? 🤔⚡

📌 A proton moves at

$2.0 \times 10^5$

 m/s. Find its de Broglie wavelength.
📌 Find the wavelength of a 5 eV photon.

💡 Hint: Use

$\lambda = h / mv$

 for particles and

$\lambda = hc / E$

 for photons!

Activities:

• Numerical problems on wavelength calculations
• Group discussion on practical implications

Assessment:

• Problem set on wavelength determination

Diffraction

In the realm of diffraction, particles such as electrons, protons, and even larger molecules exhibit wave-like behavior when passing through narrow slits or encountering obstacles, creating interference patterns characteristic of wave propagation. This phenomenon challenges the classical notion of particles as localized entities and highlights the probabilistic nature of quantum particles, where their positions and momenta are described by wave functions that exhibit interference effects.

The diffraction of particles can be understood through the principles of wave mechanics, where the de Broglie hypothesis postulates that particles, in addition to their particle-like properties, also possess wave characteristics with wavelengths inversely proportional to their momentum. When particles with non-zero momentum encounter a diffracting obstacle, their wave functions spread out and interfere with each other, leading to the formation of diffraction patterns that exhibit peaks and troughs akin to those produced by waves.

The diffraction of particles provides experimental evidence supporting the wave-particle duality of matter, as particles exhibit behaviors traditionally associated with waves, such as interference and diffraction, when subjected to diffraction experiments. By observing the patterns formed by diffracting particles, physicists can infer the wave-like nature of matter and reconcile the seemingly contradictory properties of particles and waves in the quantum realm.

Wave-particle Duality

The concept of wave-particle duality stems from the revolutionary insights of quantum theory, where particles such as electrons, protons, and even larger entities exhibit behaviors characteristic of both particles and waves under different experimental conditions. This dual nature of matter manifests in various phenomena, such as the diffraction and interference patterns observed in particle experiments, the quantization of energy levels in atomic systems, and the probabilistic nature of particle interactions at the quantum scale.

At the heart of wave-particle duality lies the wave function, a mathematical description that encapsulates the probabilistic distribution of a particle's properties, such as position, momentum, and energy, in the quantum realm. The wave function embodies the wave-like nature of matter, exhibiting interference effects, diffraction patterns, and superposition states that defy classical intuitions and underscore the probabilistic nature of quantum entities.

When matter is probed experimentally, its wave-like behavior becomes evident in phenomena such as the double-slit experiment, where particles exhibit interference patterns akin to those produced by waves passing through narrow openings. This observation highlights the wave-like propagation of matter and reinforces the notion that particles possess both particle-like and wave-like attributes, depending on the experimental context and the nature of the measurement.

de Broglie

The de Broglie wavelength, symbolized by λ, relates the momentum of a particle to its wavelength, expressing the wave-like properties inherent in matter. The de Broglie wavelength of a particle is given by the equation:

$\lambda = \frac{h}{p}$

Where:
λ = De Broglie wavelength
h = Planck's constant (6.626 x 10^-34 J s)
p = Momentum of the particle

To make calculations using the de Broglie wavelength for particles, one must first determine the momentum of the particle, typically given by:

$p = mv$

Where:
m = Mass of the particle
v = Velocity of the particle

Once the momentum is calculated, one can then use the de Broglie wavelength equation to find the wavelength associated with the particle's motion. The de Broglie wavelength provides insights into the wave-like behavior of particles, allowing physicists to analyze diffraction, interference, and other quantum phenomena that manifest in the behavior of matter at the subatomic level.