Electron Photon Same de Broglie Wavelength Comparison

The de Broglie wavelength ($\lambda$) of a particle is fundamentally linked to its linear momentum ($p$) through the equation:

$ \lambda = \frac{h}{p} $

Here, $h$ represents Planck's constant.

Equating Wavelengths

The question states that an electron and a photon possess the same de Broglie wavelength. Based on the de Broglie relation ($\lambda = h/p$), this condition directly implies that their linear momenta must be equal:

$ p_{electron} = p_{photon} $

Let's denote this shared momentum value as $p$. We now examine how this equality affects other properties:

Property Analysis

• Linear Momentum: As established, $p_{electron} = p_{photon}$. They share the same linear momentum.
• Energy: A photon's energy is calculated as $E_{photon} = pc$. An electron's kinetic energy is $KE_{electron} = \frac{p^2}{2m_{electron}}$. Due to the electron's non-zero mass ($m_{electron}$), these energies differ.
• Speed: Photons always travel at the speed of light, $c$. The electron's speed is $v_{electron} = \frac{p}{m_{electron}}$. Since $p = p_{photon}$, $v_{electron} = \frac{p_{photon}}{m_{electron}}$. Given that $m_{electron}$ is finite and non-zero, $v_{electron}$ cannot equal $c$. Therefore, their speeds are not the same.
• Angular Momentum: Angular momentum ($L$) depends on linear momentum ($p$), position vector ($\vec{r}$), and the chosen reference point. For a particle moving rectilinearly, its orbital angular momentum magnitude is $L = r_\perp p$, where $r_\perp$ is the perpendicular distance from the reference point to the path. If we make the specific assumption that the electron and photon travel along paths that are the same perpendicular distance ($r_\perp$) from a common reference point, and their momenta align, then their equal linear momenta ($p$) result in equal angular momenta ($L = r_\perp p$). This scenario provides a basis for equating their angular momenta.

Result

Based on the direct implication of equal linear momentum derived from equal de Broglie wavelengths, and requiring the specific assumption regarding trajectories to equate orbital angular momentum, the electron and photon have the same angular momentum.