Protons are being accelerated in a particle accelerator. When the energy of the protons is doubled, their de Broglie wavelength will
A) decrease by a factor of 2 .
B) decrease by a factor of $\sqrt { 2 }$.
C) increase by a factor of 2 .
D) increase by a factor of 4 .
E) increase by a factor of $\sqrt { 2 }$.

Question

Quizplus · Accepted Answer

The de Broglie wavelength $\lambda$ of a particle is given by $\lambda = \frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum of the particle. For a proton in a particle accelerator, its kinetic energy $E$ is related to its momentum $p$ by the relativistic relation $E^2 = p^2c^2 + m^2c^4$, where $m$ is the rest mass of the proton and $c$ is the speed of light. When the energy of the protons is doubled, the increase in momentum is less than double due to the relativistic effect, leading to a decrease in the de Broglie wavelength. However, without delving into complex relativistic calculations, a simpler non-relativistic approximation can be used for understanding the trend: $E \propto p^2$, so if $E$ is doubled, $p$ increases by a factor of $\sqrt{2}$, and thus $\lambda$ decreases by a factor of $\sqrt{2}$, since $\lambda$ is inversely proportional to $p$.

Protons Are Being Accelerated in a Particle Accelerator 2\sqrt { 2 }2​

Protons Are Being Accelerated in a Particle Accelerator $\sqrt { 2 }$