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 determine the correct graphical representation of the relationship between de Broglie wavelength \( \lambda \) and kinetic energy \( K \) for a particle of constant mass, we start with the de Broglie wavelength formula: \(\lambda = \frac{h}{p}\), where \( h \) is Planck's constant and \( p \) is momentum. The momentum \( p \) of a particle is given by \(\sqrt{2mK}\) for a particle with mass \( m \) and kinetic energy \( K \). Substituting gives:

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

This shows an inverse relationship between \(\lambda\) and \(\sqrt{K}\). Squaring both sides results in:

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

Graphing \(\lambda^2\) vs \(K\) will yield a hyperbolic curve. 
The correct representation is which accurately depicts \(\lambda^2\) decreasing as \(K\) increases, in line with the inverse proportionality.