A photon and an electron have equal energy `E . lambda_("photon")//lambda_("electron")` is proportional to

Options

A. `sqrt( E)`
B. `1//sqrt( E)`
C. `1//E`
D. Does not depend upon `E`

Explanation

To solve the problem of finding the proportional relationship between the wavelengths of a photon and an electron when both have equal energy \( E \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy of a Photon**:
The energy of a photon is given by the equation:
\[ E = \frac{hc}{\lambda_{\text{photon}}} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda_{\text{photon}} \) is the wavelength of the photon.

2. **Understanding the Energy of an Electron**:
The energy of a relativistic electron can be expressed as:
\[ E = \sqrt{(pc)^2 + (m_0c^2)^2} \]
where \( p \) is the momentum of the electron and \( m_0 \) is the rest mass of the electron.

3. **Relating Momentum to Wavelength**:
The de Broglie wavelength \( \lambda \) of a particle is given by:
\[ \lambda = \frac{h}{p} \]
Therefore, we can express the wavelengths of the photon and the electron in terms of their momenta.

4. **Finding the Momentum of the Photon**:
The momentum of a photon is given by:
\[ p_{\text{photon}} = \frac{E}{c} \]

5. **Finding the Momentum of the Electron**:
For a non-relativistic electron, the momentum can be approximated as:
\[ p_{\text{electron}} = \sqrt{2m_0E} \]
where \( m_0 \) is the rest mass of the electron.

6. **Expressing Wavelengths**:
Using the relationships for momentum, we can express the wavelengths as:
\[ \lambda_{\text{photon}} = \frac{h}{p_{\text{photon}}} = \frac{hc}{E} \]
\[ \lambda_{\text{electron}} = \frac{h}{p_{\text{electron}}} = \frac{h}{\sqrt{2m_0E}} \]

7. **Finding the Ratio of Wavelengths**:
Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_{\text{photon}}}{\lambda_{\text{electron}}} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2m_0E}}} \]
Simplifying this gives:
\[ \frac{\lambda_{\text{photon}}}{\lambda_{\text{electron}}} = \frac{c \sqrt{2m_0E}}{E} \]

8. **Final Proportional Relationship**:
Since \( E \) is the same for both the photon and the electron, we can conclude that:
\[ \frac{\lambda_{\text{photon}}}{\lambda_{\text{electron}}} \propto \frac{c \sqrt{2m_0}}{\sqrt{E}} \]
Thus, we can say:
\[ \lambda_{\text{photon}} \propto \frac{1}{\sqrt{E}} \]

### Conclusion:
The ratio \( \frac{\lambda_{\text{photon}}}{\lambda_{\text{electron}}} \) is proportional to \( \frac{1}{\sqrt{E}} \).