Which of the following figure represents the variation of particle momentum and the associated de - Broglie wavelength ?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which figure represents the variation of particle momentum and the associated de Broglie wavelength, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The de Broglie wavelength (λ) is related to the momentum (p) of a particle by the equation:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant.

2. **Identify the Inverse Relationship**: From the equation, we can see that:
\[ \lambda \propto \frac{1}{p} \]
This means that as momentum (p) increases, the de Broglie wavelength (λ) decreases, and vice versa.

3. **Rearrange the Equation**: We can also express momentum in terms of wavelength:
\[ p = \frac{h}{\lambda} \]
This shows that momentum is inversely proportional to the wavelength.

4. **Differentiate the Relationship**: To analyze the nature of the graph, differentiate the equation \( p = \frac{h}{\lambda} \):
\[ \frac{dp}{d\lambda} = -\frac{h}{\lambda^2} \]
The negative sign indicates that as λ increases, p decreases, which suggests a downward curve.

5. **Analyze the Graphs**: Now we need to evaluate the provided graphs:
- **Graph 1**: Linear variation - not possible since the relationship is not linear.
- **Graph 2**: Curved but λ increases while p also increases - incorrect.
- **Graph 3**: Linear graph - again not possible.
- **Graph 4**: Curved graph where λ increases and p decreases - this fits our derived relationship.

6. **Conclusion**: The fourth graph correctly represents the inverse relationship between particle momentum and the associated de Broglie wavelength.

### Final Answer:
The correct figure that represents the variation of particle momentum and the associated de Broglie wavelength is **Graph 4**.