Principles of Modern Physics: Relativity and Quantum Theory

Frames of Reference

Inertial frame of reference – A frame at rest or moving with constant velocity where Newton’s First Law of Motion holds true and no extra forces are needed.

Non-inertial frame of reference – A frame that is accelerating or rotating where Newton’s laws need fictitious (pseudo) forces to explain motion.

Radiation Laws and Hypotheses

Wien’s displacement law – It states that the wavelength at which a body emits maximum radiation is inversely proportional to its temperature. λmax ∝ 1/T and λmaxT = Constant.

Stefan’s law – It states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of its absolute temperature.

de Broglie hypothesis – It states that every moving particle is associated with a wave, called a matter wave. This means particles like electrons show both particle and wave nature. λ = h / p = h / mv

Proper Length – The proper length of an object is the length measured in the frame of reference in which the object is at rest.

Postulates of Special Relativity

1. Principle of Relativity – The laws of physics are the same in all inertial frames of reference (no frame is special).
2. Constancy of Speed of Light – The speed of light in a vacuum is always constant for all observers, regardless of the motion of the source or observer.

Planck’s Quantum Hypothesis

It states that energy is not emitted or absorbed continuously, but in the form of small discrete packets called quanta. The energy of each quantum is directly proportional to the frequency of radiation: E = hν.

Assumptions of Planck’s Hypothesis

• Energy is quantized: Energy is emitted or absorbed in discrete packets, not continuously.
• Energy–frequency relation: The energy of each quantum is E = hν.
• Emission/absorption in whole quanta: Energy is emitted or absorbed only in integral multiples of hν.
• Energy depends on frequency: The energy of radiation depends only on its frequency, not on its intensity.

Planck’s radiation formula – It gives the energy distribution of radiation emitted by a black body as a function of frequency and temperature.

COMPTON EFFECT

Black Body Physics

Black body – An ideal object that absorbs all radiation incident on it, regardless of wavelength or angle, and reflects or transmits none.

Black body radiation – The radiation emitted by a black body due to its temperature. It has a continuous spectrum that depends only on the temperature of the body, not on its material.

Black body spectrum – The plot of the intensity of radiation emitted by a black body versus wavelength at a given temperature.

Spectrum Characteristics

• Continuous Spectrum: Radiation is emitted at all wavelengths.
• Temperature Dependence: As temperature increases, total energy emitted increases (Stefan-Boltzmann law) and peak wavelength shifts toward shorter wavelengths (Wien’s displacement law).
• Shape of Curve: The spectrum rises sharply to a maximum and then gradually decreases.
• Maximum Intensity: Occurs at a wavelength λmax which depends on temperature.
• Explanation: Max Planck resolved the ultraviolet catastrophe using quantized energy levels.

Phase and Group Velocity

Michelson-Morley Experiment

The Michelson-Morley experiment (1887) was designed to detect the presence of aether, a hypothetical medium through which light was thought to propagate. Its failure was crucial for the development of Einstein’s theory of special relativity.

Experimental Arrangement

1. Light source: Typically monochromatic light.
2. Half-silvered mirror: Splits the light beam into two perpendicular paths.
3. Two mirrors (M1 and M2): Reflect the beams back toward the splitter.
4. Telescope: To observe the interference pattern.
5. Rotating platform: To check for changes in interference fringes.

Working Principle

• Splitting: The beam hits the half-silvered mirror; half reflects to M1, half passes to M2.
• Reflection: Both beams reflect and return to the splitter.
• Interference: Recombined beams produce fringe shifts if travel times differ.
• Rotation: The apparatus is rotated 90° to test for fringe shifts.

Observations and Conclusion

• Observation: No significant shift in interference fringes was detected.
• Conclusion: The speed of light is constant in all directions, disproving the aether hypothesis.

Significance of the Negative Result

• Disproved the existence of aether.
• Supported the constancy of light speed in all inertial frames.
• Provided a basis for the Special Theory of Relativity.

Einstein Mass-Energy Relation

The mass–energy relation states that mass and energy are equivalent. The energy E of a body at rest is related to its rest mass m by: E = mc², where c is the speed of light in a vacuum.

Practice Problems

Question: Estimate the total radiation emitted per second by the Sun, assuming it radiates as a blackbody with surface temperature 1700°C and radius 7×10⁸ m.

Question: Compare the temperature of the Sun and the Moon, given that the wavelength of maximum radiation emitted from the Sun and the Moon are 4753 Å and 14 μm, respectively.

Question: A particle of charge q and mass m is accelerated through a potential difference V. Find its de Broglie wavelength. Calculate λ if the particle is an electron and V=50V.