Understanding de-Broglie Wavelength
The de-Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like nature of particles. It provides a relationship between the wavelength of a particle and its momentum.
Key Relationship
- The formula for the de-Broglie wavelength (λ) is given by:
- λ = h/p
- Here, h is Planck's constant, and p is the momentum of the particle.
Momentum Definition
- Momentum (p) of an object is defined as:
- p = mv
- Where m is the mass and v is the velocity of the object.
Proportionality of Wavelength
- As per the de-Broglie hypothesis:
- λ ∝ 1/p
- This implies that the wavelength is inversely proportional to the momentum.
Why Option C is Correct
- Given that:
- p = mv (mass times velocity), an increase in momentum results in a decrease in the wavelength.
- Therefore, when momentum increases, the de-Broglie wavelength decreases, which aligns with option 'C' (λ ∝ 1/p).
Other Options Explained
- a) λ ∝ 1/V: Incorrect as velocity alone does not define wavelength.
- b) λ ∝ 1/m: While mass affects momentum, it doesn’t directly correlate to wavelength without considering velocity.
- d) λ ∝ p: This is incorrect; it's the inverse relationship that holds true.
Conclusion
- The correct answer to the de-Broglie wavelength proportionality is indeed option 'C', emphasizing the inverse relationship between wavelength and momentum. This understanding is crucial for comprehending wave-particle duality in quantum mechanics, especially in the context of JEE syllabus.