Example 13 Find the ratio of de Broglie wavelengths of two particles with the same momenta but different charges (q_(1)=2q_(2))

To find the ratio of de Broglie wavelengths of two particles with the same momenta but different charges, we can use the de Broglie wavelength formula:<br /><br />\[ \lambda = \frac{h}{p} \]<br /><br />where \( \lambda \) is the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle.<br /><br />Given that the momenta of the two particles are the same, we can denote the momentum as \( p \).<br /><br />For the first particle with charge \( q_1 \), the de Broglie wavelength is:<br /><br />\[ \lambda_1 = \frac{h}{p} \]<br /><br />For the second particle with charge \( q_2 \), the de Broglie wavelength is:<br /><br />\[ \lambda_2 = \frac{h}{p} \]<br /><br />Since \( q_1 = 2q_2 \), we can express \( q_2 \) in terms of \( q_1 \):<br /><br />\[ q_2 = \frac{q_1}{2} \]<br /><br />Now, we can find the ratio of the de Broglie wavelengths:<br /><br />\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{p}}{\frac{h}{p}} = 1 \]<br /><br />Therefore, the ratio of the de Broglie wavelengths of the two particles with the same momenta but different charges is 1:1.