The wavelength `lambda_(e )` of an electron and `lambda_(p)` of a photon of same energy `E` are related by

Options

A. (a)`lambda_(p) " " alpha " " lambda ^(2)e`
B. (b) `lambda_(p) " " alpha " " lambda_(e)`
C. (c) `lambda_(p) " " alpha " " sqrt(lambda_(e))`
D. (d) `lambda_(p) " " alpha " " 1/sqrt(lambda_(e))`

Answer

To find the relationship between the wavelengths of an electron (λ_e) and a photon (λ_p) of the same energy (E), we can follow these steps:

### Step-by-Step Solution:

1. **Wavelength of an Electron:**
The wavelength of an electron can be expressed using the de Broglie wavelength formula:
\[ \lambda_e = \frac{h}{\sqrt{2mE}} \]
where:
- \( h \) is Planck's constant,
- \( m \) is the mass of the electron,
- \( E \) is the energy of the electron.

2. **Wavelength of a Photon:**
The wavelength of a photon is given by:
\[ \lambda_p = \frac{hc}{E} \]
where:
- \( c \) is the speed of light.

3. **Equating the Energy:**
Since both the electron and the photon have the same energy \( E \), we can express \( E \) from the photon wavelength equation:
\[ E = \frac{hc}{\lambda_p} \]

4. **Substituting Energy in Electron Wavelength:**
Now, substitute \( E \) from the photon equation into the electron wavelength equation:
\[ \lambda_e = \frac{h}{\sqrt{2m \left(\frac{hc}{\lambda_p}\right)}} \]
Simplifying this gives:
\[ \lambda_e = \frac{h}{\sqrt{\frac{2mhc}{\lambda_p}}} \]
\[ \lambda_e = \frac{h \sqrt{\lambda_p}}{\sqrt{2mhc}} \]

5. **Squaring Both Sides:**
To eliminate the square root, we square both sides:
\[ \lambda_e^2 = \frac{h^2 \lambda_p}{2mhc} \]

6. **Rearranging the Equation:**
Rearranging gives us:
\[ \lambda_e^2 \propto \lambda_p \]
This indicates that the square of the wavelength of the electron is directly proportional to the wavelength of the photon.

### Final Relation:
Thus, we can conclude that:
\[ \lambda_e^2 = \frac{h^2}{2mc} \lambda_p \]
This shows the relationship between the wavelengths of the electron and the photon of the same energy.