Two particles A and B have de-Broglie's wavelength `30Å` combined to from a single particle C. Momentum is conserved in this processs. The possible de-Broglile's wavelength of C is (the motion in one dimensional)

Options

A. `10Å`
B. `20Å`
C. `60Å`
D. `80Å`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Identify the initial conditions
We have two particles, A and B, each with a de Broglie wavelength of \( 30 \, \text{Å} \). We can denote the momentum of particles A and B as \( p_A \) and \( p_B \) respectively.

### Step 3: Calculate the initial momentum
Using the de Broglie wavelength formula, we can express the momentum of particles A and B:
\[ p_A = \frac{h}{\lambda_A} = \frac{h}{30 \, \text{Å}} \]
\[ p_B = \frac{h}{\lambda_B} = \frac{h}{30 \, \text{Å}} \]

Since both particles have the same wavelength, their momenta are equal:
\[ p_A = p_B \]

### Step 4: Calculate total initial momentum
The total initial momentum \( p_i \) of the system (particles A and B) is:
\[ p_i = p_A + p_B = \frac{h}{30 \, \text{Å}} + \frac{h}{30 \, \text{Å}} = \frac{2h}{30 \, \text{Å}} = \frac{h}{15 \, \text{Å}} \]

### Step 5: Apply conservation of momentum
According to the conservation of momentum, the total momentum before the combination (particles A and B) must equal the momentum after the combination (particle C):
\[ p_i = p_f \]
where \( p_f \) is the momentum of particle C.

### Step 6: Relate the momentum of particle C to its wavelength
Using the de Broglie wavelength formula for particle C:
\[ p_f = \frac{h}{\lambda_C} \]

### Step 7: Set up the equation
Since \( p_i = p_f \), we can equate the two expressions:
\[ \frac{h}{15 \, \text{Å}} = \frac{h}{\lambda_C} \]

### Step 8: Solve for the wavelength of particle C
By canceling \( h \) from both sides, we have:
\[ \lambda_C = 15 \, \text{Å} \]

### Conclusion
Thus, the possible de Broglie wavelength of particle C is \( 15 \, \text{Å} \).

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