The de Broglie wavelength associated with particle is

Options

A. inversely proportional to its momentum
B. inversely proportional to its energy
C. directly proportional to its velocity
D. directly proportional to its momentum

Answer

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

### Step 1: Understand the Concept of de Broglie Wavelength
The de Broglie wavelength (\(\lambda\)) is a concept that describes the wave-like behavior of particles. According to de Broglie, every moving particle or object has an associated wavelength.

### Step 2: Write the Formula for de Broglie Wavelength
The formula for the de Broglie wavelength is given by:
\[ \lambda = \frac{h}{p} \]
Where:
- \(\lambda\) = de Broglie wavelength
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
- \(p\) = momentum of the particle

### Step 3: Relate Momentum to Mass and Velocity
Momentum (\(p\)) can be expressed in terms of mass (\(m\)) and velocity (\(v\)):
\[ p = m \cdot v \]
Substituting this into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{m \cdot v} \]

### Step 4: Analyze the Relationship
From the equation \(\lambda = \frac{h}{m \cdot v}\), we can deduce several relationships:
- The wavelength (\(\lambda\)) is inversely proportional to momentum (\(p\)).
- The wavelength is also inversely proportional to velocity (\(v\)).
- The wavelength is not directly proportional to energy.

### Step 5: Evaluate the Options
Based on the relationships derived:
- The statement that wavelength is inversely proportional to momentum is correct.
- The statement that wavelength is inversely proportional to energy is incorrect.
- The statement that wavelength is directly proportional to velocity is incorrect, as it is inversely proportional.
- The statement that wavelength is directly proportional to momentum is also incorrect.

### Conclusion
The correct option is that the de Broglie wavelength is inversely proportional to momentum.

### Final Answer
The correct option is **A**: Wavelength is inversely proportional to momentum.

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