The de Broglie wavelength is given by

Options

A. `p=(2pih)/(lamda)`
B. `p=(h)/(2lamda)`
C. `p=(2pi)/(hlamda)`
D. `p=(2pi)/(lamda)`

Explanation

To derive the de Broglie wavelength, we start with the fundamental relationship between a particle's momentum and its wavelength. The de Broglie wavelength (\( \lambda \)) is given by the equation:

\[ \lambda = \frac{h}{p} \]

where:
- \( \lambda \) is the de Broglie wavelength,
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( p \) is the momentum of the particle.

### Step 1: Understand the Concept of Momentum
Momentum (\( p \)) of a particle is defined as the product of its mass (\( m \)) and its velocity (\( v \)):
\[ p = mv \]

### Step 2: Substitute Momentum in the De Broglie Equation
Now, we can substitute the expression for momentum into the de Broglie wavelength equation:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Interpret the Result
This equation shows that the de Broglie wavelength is inversely proportional to both the mass and the velocity of the particle. This means that as the mass or velocity of the particle increases, the wavelength decreases, indicating a wave-particle duality.

### Step 4: Alternative Form of the Equation
We can also express momentum in terms of wavelength:
\[ p = \frac{h}{\lambda} \]

This shows the relationship between the momentum of a particle and its wavelength.

### Summary
Thus, the de Broglie wavelength can be expressed in two forms:
1. \( \lambda = \frac{h}{p} \)
2. \( \lambda = \frac{h}{mv} \)