Which of the following graphs correctly represents the variation of particle momentum with associated de Broglie wavelength?

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the variation of particle momentum with associated de Broglie wavelength, we need to understand the relationship between momentum (p) and wavelength (λ) as described by the de Broglie hypothesis.

### Step-by-step Solution:

1. **Understand de Broglie's Relation**: According to de Broglie’s hypothesis, the momentum (p) of a particle is inversely proportional to its wavelength (λ). This relationship can be expressed mathematically as:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant.

2. **Rearranging the Equation**: From the above equation, we can rearrange it to show the relationship between momentum and wavelength:
\[ \lambda = \frac{h}{p} \]
This indicates that as the momentum increases, the wavelength decreases, and vice versa.

3. **Graphical Representation**: The relationship \( p = \frac{h}{\lambda} \) can be interpreted as a hyperbolic relationship. If we plot momentum (p) on the y-axis and wavelength (λ) on the x-axis, we can see that the graph will be a rectangular hyperbola, as both p and λ cannot be zero.

4. **Characteristics of the Graph**: The graph will approach the axes but will never touch them, indicating that neither momentum nor wavelength can be zero. As momentum increases, the wavelength decreases, creating the hyperbolic shape.

5. **Conclusion**: The correct graph representing the variation of particle momentum with associated de Broglie wavelength is a rectangular hyperbola.