Question

Which of the following represents de Broglie equation?

• A. \(\lambda = \frac{h}{\sqrt{mv}}\)
• B. \(\lambda = \frac{h}{mv}\)
• C. \(\lambda = \frac{h}{mp}\)
• D. \(\lambda = \frac{\mu}{p}\)

Hint:

The de Broglie equation \(\lambda = h/p\) is fundamental. Since p=mv, the two forms \(\lambda = h/p\) and \(\lambda = h/mv\) are equivalent and frequently used. Remember that this equation links a particle property (momentum) to a wave property (wavelength).

Answer 1

The Correct Option is B

Step 1: Understanding the de Broglie Hypothesis.
In 1924, Louis de Broglie proposed that all matter exhibits wave-like properties. He suggested that a particle with momentum (p) has an associated wavelength (\(\lambda\)). This is known as the wave-particle duality of matter.
Step 2: The de Broglie Equation.
The equation that relates the wavelength (\(\lambda\)) of a particle to its momentum (p) is: \[ \lambda = \frac{h}{p} \] Where:

• \(\lambda\) is the de Broglie wavelength.
• h is Planck's constant (6.626 \(\times\) 10\(^{-34}\) J·s).
• p is the momentum of the particle.

Step 3: Expressing Momentum.
Momentum (p) of a particle is defined as the product of its mass (m) and its velocity (v): \[ p = mv \] Step 4: Substituting Momentum in the de Broglie Equation.
By substituting the expression for momentum into the de Broglie equation, we get the most common form of the equation: \[ \lambda = \frac{h}{mv} \] Step 5: Evaluating the Options.

• (A) \(\lambda = \frac{h}{\sqrt{mv}}\): Incorrect.
• (B) \(\lambda = \frac{h}{mv}\): Correct. This is the de Broglie equation.
• (C) \(\lambda = \frac{h}{mp}\): Incorrect. This would imply \(\lambda = \frac{h}{m(mv)} = \frac{h}{m^2v}\).
• (D) \(\lambda = \frac{\mu}{p}\): Incorrect. It uses a different symbol (\(\mu\)) instead of Planck's constant (h).

Step 6: Final Answer.
The correct representation of the de Broglie equation is \(\lambda = \frac{h}{mv}\).