A praticle of mass M at rest decays into two particle of masses `m_1` and `m_2`, having non-zero velocities. The ratio of the de Broglie wavelength of the particles `(lamda_1)/(lamda_2)` is

Options

A. `(m_1)/(m_2)`
B. `(m_2)/(m_1)`
C. 1
D. `(sqrtm_2)/(sqrtm_1)`

Explanation

To solve the problem of finding the ratio of the de Broglie wavelengths of two particles resulting from the decay of a particle of mass \( M \), we will follow these steps:

### Step 1: Understand the conservation of momentum
Since the particle of mass \( M \) is at rest before the decay, the total momentum before the decay is zero. According to the law of conservation of momentum, the total momentum after the decay must also be zero.

### Step 2: Set up the momentum equation
Let the velocities of the two particles after decay be \( v_1 \) for mass \( m_1 \) and \( v_2 \) for mass \( m_2 \). The momentum conservation equation can be written as:
\[ 0 = m_1 v_1 + m_2 v_2 \]
This implies:
\[ m_1 v_1 = -m_2 v_2 \]

### Step 3: Relate the momenta of the two particles
From the equation \( m_1 v_1 = -m_2 v_2 \), we can express the magnitudes of the momenta:
\[ |m_1 v_1| = |m_2 v_2| \]
This means that the magnitudes of the momenta of both particles are equal:
\[ |p_1| = |p_2| \]
where \( p_1 = m_1 v_1 \) and \( p_2 = m_2 v_2 \).

### Step 4: Use de Broglie's hypothesis
According to de Broglie's hypothesis, the 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.

### Step 5: Write the expressions for the wavelengths
For the two particles, we have:
\[ \lambda_1 = \frac{h}{p_1} \quad \text{and} \quad \lambda_2 = \frac{h}{p_2} \]

### Step 6: Find the ratio of the wavelengths
Now, we can find the ratio of the de Broglie wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{p_1}}{\frac{h}{p_2}} = \frac{p_2}{p_1} \]
Since we established that \( |p_1| = |p_2| \), we can substitute:
\[ \frac{\lambda_1}{\lambda_2} = \frac{|p_2|}{|p_1|} = 1 \]

### Conclusion
Thus, the ratio of the de Broglie wavelengths of the two particles is:
\[ \frac{\lambda_1}{\lambda_2} = 1 \]