The de -Brogile wavelength associated with a matter particle is

Options

A. directly proportional to the momentum of the particle
B. Directly proportional to the velocity of the particle
C. inversely proportional to the momentum of the particle
D. Inversely proportional of planck's constant

Answer

To solve the question regarding the de Broglie wavelength associated with a matter particle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the de Broglie Wavelength Formula**:
The de Broglie wavelength (λ) of a matter particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where:
- \( \lambda \) = de Broglie wavelength
- \( h \) = Planck's constant
- \( m \) = mass of the particle
- \( v \) = velocity of the particle

2. **Identifying Momentum**:
The momentum (p) of the particle is defined as:
\[ p = mv \]
Therefore, we can also express the de Broglie wavelength in terms of momentum:
\[ \lambda = \frac{h}{p} \]

3. **Analyzing Proportional Relationships**:
From the formula \( \lambda = \frac{h}{mv} \), we can deduce the following relationships:
- The de Broglie wavelength is **inversely proportional** to the momentum (p) of the particle.
- Since \( p = mv \), it is also inversely proportional to the velocity (v) of the particle.

4. **Evaluating the Options**:
Now, let's evaluate the options provided in the question:
- **Option 1**: "Directly proportional to the momentum of the particle." - **Incorrect** (it is inversely proportional).
- **Option 2**: "Directly proportional to the velocity of the particle." - **Incorrect** (it is inversely proportional).
- **Option 3**: "Inversely proportional to the momentum of the particle." - **Correct** (as derived).
- **Option 4**: "Inversely proportional to Planck's constant." - **Incorrect** (Planck's constant is a constant and does not vary).

5. **Conclusion**:
The correct answer is **Option 3**: "Inversely proportional to the momentum of the particle."