Which is the de-Broglie equation?

Options

A. (a) `h=plamda`
B. (b) `h=plamda^(-1)`
C. (c) `h=lamdap^(-1)`
D. (d) `h=p+lamda`

Answer

To derive the de Broglie equation step by step, we start with the concept of dual nature of matter and radiation.

### Step-by-Step Solution:

1. **Understanding Dual Nature**:
- Louis de Broglie proposed that particles, like electrons, exhibit both wave-like and particle-like properties. This is known as the dual nature of matter.

2. **Energy of a Photon**:
- The energy (E) of a photon can be expressed using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength.

3. **Relating Energy to Mass**:
- According to Einstein's theory, the energy of a particle with mass \( m \) can also be expressed as:
\[ E = mc^2 \]

4. **Setting the Two Energy Equations Equal**:
- Since both expressions represent energy, we can set them equal to each other:
\[ \frac{hc}{\lambda} = mc^2 \]

5. **Rearranging the Equation**:
- We can rearrange this equation to solve for the wavelength \( \lambda \):
\[ \lambda = \frac{hc}{mc^2} \]

6. **Simplifying the Equation**:
- We can simplify this further. Since \( mc \) is the momentum \( p \) of the particle (where \( p = mv \)):
\[ \lambda = \frac{h}{p} \]

7. **Final Form of the de Broglie Equation**:
- Thus, the de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
- This is the de Broglie equation, which relates the wavelength of a particle to its momentum.

### Conclusion:
The de Broglie equation is:
\[ \lambda = \frac{h}{p} \]