If wavelength of particle moment `P` is equal to `lamda`, then what will be its wavelength for momentum 1.5P

Options

A. `(2)/(3)lamda`
B. `(4)/(3)lamda`
C. `(3)/(2)lamda`
D. `lamda`

Answer

To solve the problem, we need to use the relationship between the wavelength (λ) of a particle and its momentum (P). According to de Broglie's hypothesis, the wavelength of a particle is inversely proportional to its momentum. This relationship can be expressed mathematically as:

\[ \lambda \propto \frac{1}{P} \]

This means that if the momentum increases, the wavelength decreases, and vice versa.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The relationship between wavelength (λ) and momentum (P) can be expressed as:
\[ \lambda = \frac{h}{P} \]
where \(h\) is Planck's constant.

2. **Initial Conditions**:
We are given that for a momentum \(P\), the wavelength is \(\lambda\):
\[ \lambda = \frac{h}{P} \]

3. **New Momentum**:
We need to find the wavelength when the momentum is increased to \(1.5P\).

4. **Apply the Relationship for New Momentum**:
For the new momentum \(1.5P\), the new wavelength \(\lambda'\) can be expressed as:
\[ \lambda' = \frac{h}{1.5P} \]

5. **Relate the New Wavelength to the Original Wavelength**:
We can express \(\lambda'\) in terms of \(\lambda\):
\[ \lambda' = \frac{h}{1.5P} = \frac{1}{1.5} \cdot \frac{h}{P} = \frac{1}{1.5} \cdot \lambda \]

6. **Simplify the Expression**:
\[ \lambda' = \frac{2}{3} \lambda \]

7. **Conclusion**:
The wavelength for momentum \(1.5P\) is:
\[ \lambda' = \frac{2}{3} \lambda \]

### Final Answer:
The wavelength for momentum \(1.5P\) is \(\frac{2}{3} \lambda\).