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.