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:
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The de Broglie wavelength is related to the kinetic energy, and this relationship is depicted graphically as an inverse square root dependence.
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.
which accurately depicts \(\lambda^2\) decreasing as \(K\) increases, in line with the inverse proportionality.
