Question

In which of the following situations, heavier particle has smaller De-Broglie wavelength. If two particles :
A. move with same speed
B. move with same linear momentum
C. move with same kinetic energy
D. have fallen through the same height
Choose the most appropriate answer from the options given below :

• A. A, B only
• B. A, B, C only
• C. A, C, D only
• D. A, B, D only

Hint:

Whenever evaluating proportionality questions, write down the formula that contains only the variable in question and the constant given. If the mass $m$ ends up anywhere in the denominator, the relationship is inverse, meaning "heavier = smaller".

Answer 1

The Correct Option is C

Step 1: Concept:
The question explores the relationship between mass ($m$) and the de Broglie wavelength ($\lambda$) under various constrained physical conditions. We must identify which conditions mathematically force a heavier mass to yield a smaller wavelength.

Step 2: Key Formula or Approach:
The foundational de Broglie wavelength equation is:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
where $h$ is Planck's constant, $p$ is momentum, and $v$ is velocity.
We can also express it in terms of Kinetic Energy ($K = \frac{p^2}{2m} \implies p = \sqrt{2mK}$):
\[ \lambda = \frac{h}{\sqrt{2mK}} \]

Step 3: Step-by-step Explanation:
Let's evaluate each condition:

• A. Move with same speed ($v$ is constant):
Using $\lambda = \frac{h}{mv}$, if $v$ is locked as a constant, then $\lambda \propto \frac{1}{m}$. Thus, a larger mass $m$ directly results in a smaller $\lambda$. (True)

• B. Move with same linear momentum ($p$ is constant):
Using $\lambda = \frac{h}{p}$, if $p$ is identical for both particles, then $\lambda$ is identical, regardless of their masses. The heavier particle does not have a smaller wavelength; they are exactly equal. (False)

• C. Move with same kinetic energy ($K$ is constant):
Using $\lambda = \frac{h}{\sqrt{2mK}}$, if $K$ is locked as a constant, then $\lambda \propto \frac{1}{\sqrt{m}}$. Because $m$ is in the denominator, a larger mass still results in a smaller $\lambda$. (True)

• D. Have fallen through the same height ($h_{drop}$ is constant):
When falling freely from rest, potential energy converts to kinetic energy ($m g h_{drop} = \frac{1}{2} m v^2$). Notice that mass cancels out: $v = \sqrt{2 g h_{drop}}$. Because gravity accelerates all masses equally, both particles achieve the exact same final speed $v$. As proven in scenario A, if they have the same speed, the heavier particle will have a smaller $\lambda$. (True)

Step 4: Final Answer:
The conditions where the heavier particle has a smaller wavelength are A, C, and D. This corresponds to option (C).