Question

A particle is dropped from a height '\(y\)'. The de-Broglie wavelength of the particle as a function of height is

• A. \(y\)
• B. \(y^{1/2}\)
• C. \(y^{0}\)
• D. \(y^{-1/2}\)

Hint:

Find the speed after falling through height y, write the momentum, and use the de Broglie relation to see how the wavelength depends on y.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The de Broglie wavelength is \(\lambda=\dfrac{h}{p}=\dfrac{h}{mv}\).

Step 2: Speed after falling
For a particle dropped from rest, \(v^2=2gy\), so \(v=\sqrt{2gy}\).

Step 3: Calculation
\(\lambda=\dfrac{h}{m\sqrt{2gy}}\). Here \(h\), \(m\) and \(g\) are constants, so \(\lambda\propto y^{-1/2}\).

Final Answer:
The wavelength varies as y to the power minus one half, which is option (D). \[ \boxed{\text{(D) }y^{-1/2}} \]