Question

A photon, an electron and a uranium nucleus all have the same wavelength. The one with the most energy:
A. Is the photon
B. Is the electron
C. Is the uranium nucleus
D. Depends upon the wavelength and the properties of the particle.

Answer 1

Hint: The energy can be expressed in terms of momentum. This momentum is related to wavelength through the de Brogle relation. Louis de Broglie was the first one to propose the wave particle duality and gave the following relation:
     \[^{\lambda = \dfrac{h}{p} = \dfrac{h}{{\sqrt {2mE} }}}\]

Complete step by step answer:
According to Quantum Mechanics, any matter (small particle or big particle) can exhibit both particle and wave nature. The behaviour of anything in the universe can be interpreted in particle form as well as waveform.
The most famous example is that of light and electrons. Light was originally thought of as waves but Newton suggested that a particle can only go in a straight line after refraction. Later Fresnel, Huygens and Hooke explained these phenomena based on the wave nature of light.
Now the light behaves in the form of a particle in photoelectric effect while as a wave in diffraction, interference etc. Same goes with electrons.
Louis de Broglie first gave the matter wave hypothesis in 1924 in which he stated that not only light (photons) but anything in the universe can exhibit both the forms and gave the following relation:
\[
  \lambda = \dfrac{h}{p} = \dfrac{h}{{\sqrt {2mE} }} \\
  E = \dfrac{{{h^2}}}{{2m{\lambda ^2}}} \\
 \]
So, according to the relation derived above, the lesser the mass the more will be the energy. Therefore, Uranium will have minimum energy while photons will have maximum energy, wavelength remains the same.
The correct answer is option A.

Note:The mass of the photon is divided into two categories because of its speed. The inertial mass or the rest mass and the relativistic mass or moving mass. Due to the speed of photons i.e. the speed of light it has zero rest mass. This means that the energy will be infinite. Well this is not the case, thanks to Einstein. Here for comparison we have considered the moving mass. $E = m{c^2}$