A photon and an electron (mass m) have the same energy E. The ratio (λ_photon / λ_electron) of their de Broglie wavelengths is (c is the speed of light):

(1) √E / 2m
(2) c√(2mE)
(3) c√(2m/E)
(4) (1/c) √(E/2m)

To find the ratio of the de Broglie wavelength of a photon to that of an electron when both have the same energy $E$, we derive the wavelength expression for each particle step-by-step.

Step 1: de Broglie Wavelength of the Photon (${\lambda }_{\text{photon}}$)
For a photon, the relationship between energy $E$ and wavelength is given by the Planck-Einstein relation:
$E=\frac{hc}{{\lambda }_{\text{photon}}}$
where $h$ is Planck's constant and $c$ is the speed of light.
Rearranging this formula for ${\lambda }_{\text{photon}}$ gives:
${\lambda }_{\text{photon}}=\frac{hc}{E}$

Step 2: de Broglie Wavelength of the Electron (${\lambda }_{\text{electron}}$)
The de Broglie wavelength of a particle with mass $m$ and momentum $p$ is:
${\lambda }_{\text{electron}}=\frac{h}{p}$
The kinetic energy $E$ of a non-relativistic electron is related to its momentum $p$ by:
$E=\frac{p^2}{2m}\Rightarrow p=\sqrt{2mE}$
Substituting this momentum back into the wavelength formula yields:
${\lambda }_{\text{electron}}=\frac{h}{\sqrt{2mE}}$

Step 3: Calculating the Ratio ($\frac{{\lambda }_{\text{photon}}}{{\lambda }_{\text{electron}}}$)
Now we divide the wavelength of the photon by the wavelength of the electron:
$\frac{{\lambda }_{\text{photon}}}{{\lambda }_{\text{electron}}}=\frac{\left(\frac{hc}{E}\right)}{\left(\frac{h}{\sqrt{2mE}}\right)}$
Simplifying the expression by cancelling the common term $h$ in the numerator and denominator:
$\frac{{\lambda }_{\text{photon}}}{{\lambda }_{\text{electron}}}=\frac{c\sqrt{2mE}}{E}$
We can rewrite the energy $E$ in the denominator as $\sqrt{E}\cdot \sqrt{E}$ to simplify the square root:
$\frac{{\lambda }_{\text{photon}}}{{\lambda }_{\text{electron}}}=c\sqrt{\frac{2mE}{E^2}}=c\sqrt{\frac{2m}{E}}$

Therefore, the ratio of their wavelengths is $c\sqrt{\frac{2m}{E}}$, which corresponds to Option (3).