Question

A particle of mass M at rest decays into two masses ${m_1}$ and ${m_2}$ with non-zero velocities. The ratio ${\lambda _1}/{\lambda _2}$ of de Broglie wavelengths of particles is
A. ${m_2}/{m_1}$
B. ${m_1}/{m_2}$
C. $\sqrt {{m_2}} /\sqrt {{m_1}}$
D. $1.0$

Answer 1

According to de-Broglie’s hypothesis, the momentum of the particle is inversely proportional to the wavelength of the particle. Use the law of conservation of momentum to relate the momentum of both particles using de-Broglie’s hypothesis.

Formula used:
$p = \dfrac{{hc}}{\lambda }$
Here, h is Planck’s constant, c is the speed of light and $\lambda$ is the de Broglie wavelength.
The linear momentum of the particle of mass m moving with velocity v is,
$p = mv$

Complete step by step answer:
We know that, according to de-Broglie’s hypothesis, the momentum of the particle is,
$p = \dfrac{{hc}}{\lambda }$
Here, h is Planck’s constant, c is the speed of light and $\lambda$ is the de Broglie wavelength.

According to the law of conservation of momentum, the momentum of a system remains conserved.
Therefore, we can write,
$Mv = {m_1}{v_1} + {m_2}{v_2}$
Here, v is the velocity of parent particle, ${v_1}$ is the velocity ${m_1}$ and ${m_1}$ is the velocity of ${m_2}$. Since the parent particle is at rest, the initial velocity v is zero. Therefore, the above equation becomes,
$0 = {m_1}{v_1} + {m_2}{v_2}$
$\Rightarrow {m_1}{v_1} = - {m_2}{v_2}$
Therefore, from the above equation, the momentum of the particle of mass ${m_1}$ and the momentum of the particle of mass ${m_2}$ is equal.

So, we can write,
${p_1} = {p_2}$
$\Rightarrow \dfrac{{hc}}{{{\lambda _1}}} = \dfrac{{hc}}{{{\lambda _2}}}$
Planck’s constant h and speed of light c is constant for both particles. Therefore, the wavelength of these particles is the same. Therefore, we can write,
$\therefore \dfrac{{{\lambda _1}}}{{{\lambda _2}}} = 1$

So, the correct answer is option (D).

Note: The negative sign for ${v_2}$ implies that the motion of the second particle is opposite to the direction of the first particle. While solving these types of questions, students can blindly use the law of conservation of momentum. Since the Planck’s constant and speed of light is constant, you can directly take the ratio of wavelength by de-Broglie hypothesis.