Quantum physics-1

Corpuscular Theory vs Wave Theory

• Corpuscular Theory of Light: Proposed by Newton; fails to explain phenomena like interference, diffraction, and polarization.

• Wave Theory of Light: Proposed by Huygens; successfully explains the aforementioned phenomena but not newer ones like the Compton effect.

• Emergence of Quantum Theory: Limitations of classical theories led to the development of quantum theory, particularly at the beginning of the 20th century.

Black Body Radiation

• Definition: A perfect black body absorbs all incident radiation at all wavelengths and emits radiation when heated.

• Black Body Characteristics: Emits full radiation after reaching thermal equilibrium; emissions depend on temperature, not the nature of material.

• Examples: Copper sphere coated with lamp black demonstrating black body properties.

Energy Distribution in Black Body Radiation

• Key Observations:

  • Energy distribution is not uniform; varies with temperature.

  • Intensity (E) increases with wavelength to a maximum at a specific wavelength (λmax) then decreases.

  • As temperature rises, λmax shifts towards shorter wavelengths.

  • Total energy emitted is proportional to the area under the intensity curve in its graph.

Laws of Black Body Radiation

1. Stefan-Boltzmann Law

• Statement: Total radiant energy (E) is proportional to the fourth power of temperature (T).

• Formula: E ∝ T^4

• Includes Stefan constant (σ): E = σT^4.

2. Wien's Displacement Law

• Statement: Product of wavelength (λmax) at peak energy and absolute temperature (T) is a constant.

• Formula: λmax * T = constant.

3. Rayleigh-Jeans Law

• Statement: Energy distribution is proportional to absolute temperature and inversely proportional to the fourth power of wavelength.

• Formula: E ∝ T / λ^4.

• Results in divergence at high frequencies, failing to match experimental results.

Planck’s Quantum Theory

• Introduction: Max Planck proposed a quantum theory to explain the inadequacies of classical laws by introducing quantized energy.

• Planck's Hypothesis: Energy changes occur in discrete units called quanta. The energy (E) of a quantum is expressed as E = nhν, where n is an integer, h is Planck’s constant, and ν is frequency.

Assumptions of Planck's Theory

1. A black body contains oscillators that vibrate at all frequencies.

2. The energy emitted by an oscillator corresponds to its vibration frequency.

3. An oscillator emits energy in multiples of a quantum.

4. Exchange of energy between radiation and matter is limited to discrete values.

Planck’s Radiation Law

• Derivation leads to energy density equations related to frequency range.

• Implies consistent outcomes aligned with empirical data.

De Broglie's Concept of Matter Waves

• Hypothesis: Suggests that matter exhibits both particle and wave properties.

• Formulation: λ = h/p, where λ is de Broglie wavelength, h is Planck's constant, and p is the momentum of the particle.

Schrödinger Wave Equations

Time-independent equation

• Describes particles' behavior in quantum mechanics as waves.

Time-dependent equation

• Accounts for varying potential energies and provides a more complex view of particles.

Heisenberg’s Uncertainty Principle

• States that position (x) and momentum (p) cannot be precisely measured at the same time.

• Formula: Δx * Δp ≥ h/(4π).

• Applies a similar principle to energy and time.