Question

If \( \lambda \) and \( K \) are de Broglie wavelength and kinetic energy, respectively, of a particle with constant mass. The correct graphical representation for the particle will be:

• A.
• B.
• C.
• D.

Hint:

The de Broglie wavelength is related to the kinetic energy, and this relationship is depicted graphically as an inverse square root dependence.

Answer 1

The Correct Option is A

To represent the relationship between de Broglie wavelength \( \lambda \) and kinetic energy \( K \) for a particle of constant mass, we utilize the de Broglie wavelength formula: \(\lambda = \frac{h}{p}\). Here, \( h \) is Planck's constant and \( p \) is momentum. For a particle with mass \( m \) and kinetic energy \( K \), momentum is expressed as \(\sqrt{2mK}\). Substituting this into the de Broglie equation yields:

\[\lambda = \frac{h}{\sqrt{2mK}}\]

This equation indicates an inverse relationship between \(\lambda\) and \(\sqrt{K}\). Squaring both sides of the equation leads to:

\[\lambda^2 \propto \frac{1}{K}\]

A graph plotting \(\lambda^2\) against \(K\) will exhibit a hyperbolic curve.
The correct graphical representation is , which illustrates \(\lambda^2\) decreasing as \(K\) increases, consistent with the derived inverse proportionality.