An electron with speed v and a photon with speed c have the...

An electron with speed $v$ and a photon with speed $c$ have the same de Broglie wavelength. If the kinetic energy and momentum of electrons is $E_{e}$ and $P_{c}$ and that of photon is $E_{p h}$ and $P_{p h}$ respectively, then the correct statement is (a) $\frac{E_{c}}{E_{p h}}=\frac{2 c}{v}$ (b) $\frac{E_{e}}{E_{p h}}=\frac{v}{2 c}$ (c) $\frac{P_{e}}{P_{p h}}=\frac{2 c}{v}$ (d) $\frac{P_{e}}{P_{p h}}=\frac{v}{2 c}$

An electron with speed $v$ and a photon with speed $c$ have the same de Broglie wavelength. If the kinetic energy and momentum of electrons is $E_{e}$ and $P_{c}$ and that of photon is $E_{p h}$ and $P_{p h}$ respectively, then the correct statement is
(a) $\frac{E_{c}}{E_{p h}}=\frac{2 c}{v}$
(b) $\frac{E_{e}}{E_{p h}}=\frac{v}{2 c}$
(c) $\frac{P_{e}}{P_{p h}}=\frac{2 c}{v}$
(d) $\frac{P_{e}}{P_{p h}}=\frac{v}{2 c}$

Key Concepts

de Broglie Wavelength

This concept establishes the wave?nature of particles by relating a particle’s momentum to a wavelength through the formula ? = h/p, where h is Planck’s constant and p is the momentum. It underpins the idea that all matter exhibits wave-like properties, a central tenet of quantum mechanics.

Momentum

Momentum, defined as the product of mass and velocity for non-relativistic particles, is a fundamental quantity in both classical and quantum physics. It directly determines the de Broglie wavelength of particles and is modified for massless particles like photons, where it relates to energy by p = E/c.

Kinetic Energy

Kinetic energy quantifies the energy associated with the motion of a particle. For massive particles, such as electrons traveling at speeds much less than light, it is typically expressed as (1/2)mv². This energy is crucial when contrasting the dynamic properties of particles with mass versus massless particles like photons.

Photon Energy-Momentum Relationship

For photons, which have zero rest mass, the relationship between energy and momentum is given by E = pc, linking their quantum energy (derived from their frequency via E = h?) directly to their momentum. This unique relationship distinguishes photons from particles with mass and is vital in understanding light's behavior.

Relativistic versus Non-Relativistic Dynamics

This concept differentiates the regimes of particle motion based on speed. Non-relativistic dynamics apply when a particle’s speed is much less than the speed of light, using classical formulas such as p = mv and E = (1/2)mv². In contrast, relativistic dynamics, necessary for particles like photons that travel at light speed, use relations like E = pc and incorporate the effects of special relativity.

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