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 velocity of the particle
D. Inversely proportional to Planck's constant

Answer

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

### Step 1: Understand the de Broglie Wavelength Formula
The de Broglie wavelength (λ) 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

### Step 2: Analyze the Relationship with Momentum
Momentum (p) of a particle is defined as:
\[ p = mv \]
We can substitute this into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} \]
This shows that the wavelength is inversely proportional to momentum.

### Step 3: Evaluate the Options
Now, let's evaluate the statements given in the question:

1. **Directly proportional to the momentum of the particle**:
- This is incorrect because \( \lambda \) is inversely proportional to momentum.

2. **Directly proportional to the velocity of the particle**:
- This is also incorrect because \( \lambda \) is inversely proportional to velocity.

3. **Inversely proportional to the velocity**:
- This is correct. As velocity increases, the wavelength decreases.

4. **Directly proportional to Planck's constant**:
- This is incorrect because \( \lambda \) is not dependent on \( h \) in a direct proportionality sense; it is dependent on \( h \) but inversely related to \( mv \).

### Conclusion
The correct answer is that the de Broglie wavelength is **inversely proportional to the velocity of the particle**.

### Final Answer
The correct option is: **Inversely proportional to the velocity of the particle**.

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