De Broglie Hypothesis: Wave-particle duality, de Broglie wavelength of an electron, and the Davisson-Germer experiment

📘 Topic Description

De Broglie Hypothesis states that every moving material particle is associated with a wave called a matter wave. Thus, not only light but matter also exhibits wave-particle duality.

λ = h/p

For a particle having momentum p = mv:

⭐ λ = h/mv

The de Broglie wavelength is inversely proportional to the momentum of the particle.

⚛️ Wave-Particle Duality

Light

Light behaves both as a wave and as a particle.

Wave phenomena include interference and diffraction.

Particle phenomena include photoelectric effect.

Matter

Moving particles also possess wave nature.

λ = h/p

Photon

E = hν

Matter Particle

p = h/λ

🧮 Derivation of de Broglie Wavelength

For a photon:

E = hν

Since:

ν = c/λ

Therefore:

E = hc/λ

From Einstein's mass-energy relation:

E = pc

Equating the two:

pc = hc/λ

Cancel c:

⭐ λ = h/p

For a particle of mass m moving with velocity v:

p = mv

Hence:

⭐ λ = h/mv

⚡ de Broglie Wavelength of an Electron

For an electron of mass m and velocity v:

λ = h/mv

If an electron is accelerated through potential difference V, then its kinetic energy is:

eV = ½mv²

Therefore:

v = √(2eV/m)

Substituting in λ = h/mv:

⭐ λ = h/√(2meV)

For an electron accelerated through V volts, the commonly used expression is:

⭐ λ ≈ 12.27/√V Å

This relation is extremely important for CBSE, JEE and NEET numerical problems.

📌 Important de Broglie Relations

Momentum

p = mv

Matter Wave

λ = h/p

Velocity Form

λ = h/mv

Electron through V

λ = h/√(2meV)

Electron in Å

λ ≈ 12.27/√V Å

🎯 Interactive Practical — Change Electron Speed

Electron velocity (relative)

Particle mass (relative)

🌊 Matter Wave Visualization

The graph below represents the qualitative idea that increasing particle momentum decreases its wavelength.

p ↑ → λ ↓

p ↓ → λ ↑

🔬 Davisson–Germer Experiment

The Davisson–Germer experiment provided experimental evidence for the wave nature of electrons.

• Electrons were accelerated through a known potential difference.
• The electron beam was directed towards a nickel crystal.
• Scattered electrons were detected at different angles.
• A strong intensity maximum was observed at a particular angle.
• This diffraction pattern confirmed the wave nature of electrons.

⭐ Conclusion: Electrons exhibit wave nature and possess a de Broglie wavelength.

🧪 Principle Behind Davisson–Germer Experiment

The nickel crystal acts like a diffraction grating for electrons. Constructive interference occurs when Bragg's condition is satisfied.

2d sinθ = nλ

The wavelength obtained experimentally agreed with the de Broglie wavelength calculated from electron momentum.

This agreement strongly verified de Broglie's matter-wave hypothesis.

🏆 Significance of Davisson–Germer Experiment

✔ Wave Nature

Confirmed wave behaviour of electrons.

✔ Diffraction

Electron diffraction was observed.

✔ de Broglie

Experimental wavelength agreed with λ = h/p.

✔ Quantum Physics

Strengthened the concept of wave-particle duality.

📊 Photon vs Matter Particle

Property	Photon	Material Particle
Energy	E = hν	Kinetic energy depends on motion
Momentum	p = h/λ	p = mv
Wavelength	λ = h/p	λ = h/p
Wave Nature	Yes	Yes for moving particles

📝 MCQ Practice — 15 Questions

1. de Broglie wavelength is given by:

A. λ = hp

B. λ = h/p

C. λ = p/h

D. λ = h + p

✔ Answer: B

2. For a particle of mass m and velocity v:

A. λ = hmv

B. λ = h/mv

C. λ = mv/h

D. λ = hm/v

✔ Answer: B

3. If momentum is doubled, de Broglie wavelength becomes:

A. Double

B. Half

C. Four times

D. Unchanged

✔ Answer: B

4. Matter waves are associated with:

A. Only electrons

B. Only photons

C. Moving material particles

D. Only protons

✔ Answer: C

5. Davisson–Germer experiment demonstrated:

A. Photoelectric effect

B. Electron diffraction

C. Nuclear fission

D. Compton effect

✔ Answer: B

6. de Broglie wavelength is inversely proportional to:

A. Energy

B. Momentum

C. Frequency

D. Charge

✔ Answer: B

7. An electron accelerated through V volts has wavelength proportional to:

A. V

B. √V

C. 1/√V

D. 1/V

✔ Answer: C

8. The experiment involving nickel crystal was:

A. Young's experiment

B. Davisson–Germer experiment

C. Millikan experiment

D. Rutherford experiment

✔ Answer: B

9. Matter waves were proposed by:

A. Einstein

B. Newton

C. de Broglie

D. Bohr

✔ Answer: C

10. For a stationary particle, its de Broglie wavelength is:

A. Zero

B. Infinite

C. One metre

D. Undefined only

✔ Answer: B

11. The wave nature of electrons is demonstrated by:

A. Diffraction

B. Photoelectric effect

C. Blackbody radiation

D. Nuclear decay

✔ Answer: A

12. The SI unit of de Broglie wavelength is:

A. metre

B. joule

C. watt

D. tesla

✔ Answer: A

13. If velocity of a particle increases, its wavelength:

A. Increases

B. Decreases

C. Remains same

D. Becomes infinite

✔ Answer: B

14. Electron diffraction confirms:

A. Particle nature only

B. Wave nature of matter

C. Absence of wavelength

D. Classical theory

✔ Answer: B

15. Bragg's law is:

A. nλ = 2d sinθ

B. nλ = d sinθ

C. λ = 2nd

D. λ = d/n

✔ Answer: A

🟣 Assertion–Reason — 5 Questions

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true but R is not the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

1. Assertion: Every moving material particle has a de Broglie wavelength.

Reason: Matter has wave nature.

Answer: A

2. Assertion: De Broglie wavelength decreases when momentum increases.

Reason: λ = h/p.

Answer: A

3. Assertion: Davisson–Germer experiment confirmed electron wave nature.

Reason: Electrons undergo diffraction from a crystal.

Answer: A

4. Assertion: A stationary particle has zero de Broglie wavelength.

Reason: Its momentum is zero.

Answer: D

5. Assertion: Electron wavelength decreases with accelerating voltage.

Reason: λ is proportional to 1/√V.

Answer: A

🟢 2 Marks — 6 Questions

Q1. State de Broglie's hypothesis.

Q2. Define matter waves.

Q3. Write the de Broglie wavelength equation.

Q4. Write the wavelength of an electron accelerated through V volts.

Q5. What was demonstrated by Davisson–Germer experiment?

Q6. Why is de Broglie wavelength significant for microscopic particles?

🟡 3 Marks — 6 Questions

Q1. Derive λ = h/p.

Q2. Derive λ = h/mv.

Q3. Explain wave-particle duality.

Q4. Explain the principle of Davisson–Germer experiment.

Q5. Explain the effect of momentum on de Broglie wavelength.

Q6. Derive λ = h/√(2meV) for an electron.

🟠 4 Marks — 6 Questions

Q1. Explain de Broglie's hypothesis with mathematical derivation.

Q2. Derive the expression for electron wavelength in terms of accelerating voltage.

Q3. Explain Davisson–Germer experiment.

Q4. Explain wave-particle duality of matter.

Q5. Explain how electron diffraction proves matter-wave nature.

Q6. Compare wavelength of particles having different momenta.

🔴 5 Marks — 6 Questions

Q1. State and derive de Broglie's matter-wave equation.

Q2. Explain Davisson–Germer experiment with its conclusion.

Q3. Derive the expression λ = 12.27/√V Å.

Q4. Explain wave-particle duality with suitable examples.

Q5. Explain the experimental verification of de Broglie's hypothesis.

Q6. Discuss the dependence of de Broglie wavelength on mass, velocity and accelerating potential.

🔵 6 Marks — 6 Questions

Q1. State de Broglie hypothesis and derive the wavelength of a moving particle.

Q2. Explain Davisson–Germer experiment in detail and show how it verified de Broglie's equation.

Q3. Derive the de Broglie wavelength of an electron accelerated through a potential V.

Q4. Explain wave-particle duality and its importance in quantum mechanics.

Q5. Discuss electron diffraction and the role of Bragg's law in the Davisson–Germer experiment.

Q6. Explain all important mathematical relations associated with matter waves.

🧮 Numerical Practice — 6 Questions

Q1. Calculate the de Broglie wavelength of an electron moving with velocity 2 × 10⁶ m/s.

Q2. Find the wavelength of an electron accelerated through a potential difference of 100 V.

Q3. An electron is accelerated through 400 V. Calculate its de Broglie wavelength in Å.

Q4. A particle has momentum 6.63 × 10⁻²⁴ kg m/s. Calculate its de Broglie wavelength.

Q5. If the momentum of an electron is doubled, what happens to its de Broglie wavelength?

Q6. An electron has a de Broglie wavelength of 1 Å. Calculate the approximate accelerating potential required.

🧪 Practical / Experiment Summary

1. Generate electrons using an electron gun.
2. Accelerate electrons through a known potential difference.
3. Direct the electron beam towards a nickel crystal.
4. Measure intensity of scattered electrons at different angles.
5. Observe a maximum intensity at a particular angle.
6. Use Bragg's law to calculate wavelength.
7. Compare experimental wavelength with λ = h/p.
8. Agreement verifies the wave nature of electrons.

⭐ Final conclusion: Matter particles such as electrons possess wave nature.

📌 Complete Formula Revision

de Broglie Equation

λ = h/p

Momentum

p = mv

Velocity Form

λ = h/mv

Electron Energy

eV = ½mv²

Electron Wavelength

λ = h/√(2meV)

Practical Formula

λ ≈ 12.27/√V Å

Bragg's Law

2d sinθ = nλ

🚀 One-Minute Revision

🔹 Every moving particle has a matter wave.
🔹 de Broglie wavelength = h/p.
🔹 p = mv.
🔹 λ = h/mv.
🔹 Higher momentum → smaller wavelength.
🔹 Electron accelerated through V: λ = h/√(2meV).
🔹 λ ≈ 12.27/√V Å.
🔹 Davisson–Germer experiment proved electron diffraction.
🔹 Nickel crystal acted as a diffraction crystal.
🔹 Bragg's law: 2d sinθ = nλ.
🔹 Experimental wavelength agreed with de Broglie wavelength.
🔹 Therefore electrons possess wave nature.

🔥 Important Search Keywords

De Broglie Hypothesis De Broglie Wavelength Electron Wavelength Wave Particle Duality Davisson Germer Experiment Matter Waves Electron Diffraction Bragg's Law Class 12 Physics CBSE Physics JEE Physics NEET Physics Modern Physics Quantum Physics Physics Numericals

📚 Class 12 Physics • CBSE • JEE • NEET • Modern Physics
🌊 De Broglie Hypothesis • Matter Waves • Davisson–Germer Experiment