Modern Physics

Concept: Particles with half-integral spin ( ) are known as fermions. Examples include electrons, protons, and neutrons. These particles obey the Pauli Exclusion Principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
Step 1: Identify the appropriate statistics.} The statistical distribution that describes the behavior of fermions is the Fermi–Dirac distribution.
Step 2: Write the distribution function.} where

• = energy of the state
• = chemical potential
• = Boltzmann constant
• = absolute temperature

Step 3: Compare with other distributions.}

• Maxwell–Boltzmann distribution applies to classical particles.
• Bose–Einstein distribution describes bosons (integral spin).

Thus, half-integer spin particles follow the Fermi–Dirac distribution.

Concept: According to Einstein's Special Theory of Relativity, the length of an object measured in the direction of motion appears shorter to an observer when the object moves at a velocity comparable to the speed of light. This phenomenon is known as length contraction. The relation for relativistic length contraction is where

• = proper length (length in the object's rest frame)
• = observed length
• = velocity of the object
• = speed of light

Step 1: Analyze the velocity dependence.} As approaches , Thus,
Step 2: Conclusion.} Therefore, the observed length becomes smaller than the proper length. Hence, the object appears contracted in the direction of motion.

Concept: According to the Bohr model, the electron in a hydrogen atom can occupy only certain discrete energy levels. The energy of the orbit is given by where is the principal quantum number.
Step 1: Identify the ground state.} The ground state corresponds to the lowest possible energy level, which occurs when
Step 2: Substitute into the energy formula.} Thus, the energy of the ground state of hydrogen is .

Concept: According to the de Broglie hypothesis, particles such as electrons exhibit wave-like behavior. The wavelength associated with a moving particle is given by where

• = Planck’s constant
• = Momentum of the particle

Step 1: Determine the kinetic energy of the electron.} When an electron is accelerated through a potential difference , where is the charge of the electron.
Step 2: Relate kinetic energy and momentum.} Thus,
Step 3: Solve for momentum.}
Step 4: Substitute into de Broglie equation.} Thus, the de Broglie wavelength is