Slide 1: Modern Physics - Change in interference conditions because of polarisation

  • Interference: The phenomenon of the superposition of two or more waves resulting in their mutual reinforcement or cancellation.
  • Polarisation: The process of confining the oscillation of a transverse wave to only one particular plane.
  • Change in Interference conditions due to Polarisation:
    1. In unpolarised light, the oscillations are in all possible directions perpendicular to the direction of propagation.
    2. When unpolarised light passes through a polariser, the vibrations along certain directions are absorbed.
    3. The transmitted light becomes linearly polarised, with vibrations in only one plane perpendicular to the propagation direction.

Slide 2: Change in Interference conditions due to Polarisation (Contd.)

  • Interference of Polarised Light:
    1. The interference of polarised light can be observed using a polariser and an analyser.
    2. Polariser: It selects one plane of vibration and allows light vibrating in that plane to pass through.
    3. Analyser: It further filters out light vibrating perpendicular to its axis.

Slide 3: Calculation of Intensity of Interference Pattern

  • The intensity of the interference pattern depends on the superposition of the amplitudes of the interfering waves.
  • Intensity at a point produced by two waves of equal amplitude:
    1. Constructive Interference: When the phase difference between the two waves is zero or an integer multiple of 2π.
    2. Destructive Interference: When the phase difference between the two waves is an odd multiple of π.

Slide 4: Conditions for Constructive Interference

  • For constructive interference, i.e., bright fringes,

    1. The path difference between the interfering waves should be an integral multiple of the wavelength.
      • Path Difference (δ) = mλ, where m is an integer.
    2. The phase difference should be zero.
      • Phase Difference (Δφ) = 0

  • The maximum intensity is obtained at these points.

Slide 5: Conditions for Destructive Interference

  • For destructive interference, i.e., dark fringes,

    1. The path difference between the interfering waves should be an odd multiple of half-wavelength.
      • Path Difference (δ) = (2m + 1)λ/2, where m is an integer.
    2. The phase difference should be an odd multiple of π.
      • Phase Difference (Δφ) = (2n + 1)π, where n is an integer.

  • The minimum intensity or complete darkness is observed at these points.

Slide 6: Examples - Constructive and Destructive Interference

Example 1: Constructive Interference

  • The path difference between two interfering waves is 3λ/4.
  • The phase difference is zero.
  • Calculate the condition for maximum intensity.
  • Solution: For constructive interference,
    • Path Difference (δ) = mλ, where m is an integer.
    • 3λ/4 = mλ, m = 3/4.
    • Therefore, the maximum intensity occurs when the path difference is 3λ/4.

Slide 7: Examples - Constructive and Destructive Interference (Contd.)

Example 2: Destructive Interference

  • The path difference between two interfering waves is λ/3.
  • The phase difference is π.
  • Calculate the condition for minimum intensity.
  • Solution: For destructive interference,
    • Path difference (δ) = (2m + 1)λ/2, where m is an integer.
    • λ/3 = (2m + 1)λ/2, m = -2/3.
    • Therefore, the minimum intensity occurs when the path difference is λ/3.

Slide 8: Interference due to Thin Films

  • Thin film interference occurs when light reflects from the upper and lower surfaces of a thin film.
  • This leads to the formation of bright and dark bands.
  • Conditions for constructive and destructive interference in thin films depend on the phase difference between the reflected waves.

Slide 9: Conditions for Constructive and Destructive Interference in Thin Films

  • For a thin film of thickness ’t’ and refractive index ’n':

    1. Constructive Interference:

      • Phase Difference (Δφ) = 2π(t/nλ)
      • t = mλ/(2n), where m is an integer

    2. Destructive Interference:

      • Phase Difference (Δφ) = (2m + 1)π(t/nλ)
      • t = (2m + 1)λ/(4n), where m is an integer

Slide 10: Example - Thin Film Interference

Example: Thin Film Interference

  • A film of refractive index ’n’ is placed on a glass plate. The thickness of the film is λ/4. Determine the refractive index of the film.
  • Solution: For constructive interference,
    • t = mλ/(2n), where m is an integer
    • t = λ/4, m = 1, n = λ/(4t)
    • Therefore, the refractive index of the film is λ/(4t).
  1. Interference due to Polarisation:
  • Interference can occur when two polarised light waves of the same wavelength and amplitude overlap.
  • The interference pattern depends on the relative phase difference between the two waves.
  • The superposition of the waves leads to constructive or destructive interference at different points.
  1. Interference of Linearly Polarised Light:
  • When linearly polarised light passes through a polariser, it becomes even more polarised.
  • The intensity of the polarised light can be calculated using the Malus’ Law: I = I₀cos²θ, where I₀ is the initial intensity and θ is the angle between the polarisation direction and the analyser axis.
  1. Interference of Circularly Polarised Light:
  • Circularly polarised light consists of two perpendicular components rotating in opposite directions.
  • When two circularly polarised light waves with the same frequency and amplitude interfere, they create an interference pattern known as the circular fringes.
  • The intensity of the circular fringes can be calculated using the equation: I = I₀(1 + cos Δφ), where I₀ is the average intensity and Δφ is the phase difference between the waves.
  1. Interference of Plane Polarised Light:
  • Plane polarised light has vibrations in only one plane perpendicular to the direction of propagation.
  • When two plane polarised light waves interfere, the resulting intensity pattern depends on the relative phase difference.
  • The intensity of the interference pattern can be calculated using the equation: I = I₁ + I₂ + 2√(I₁I₂)cos Δφ, where I₁ and I₂ are the intensities of the individual waves and Δφ is the phase difference.
  1. Applications of Interference in Polarisation:
  • Polarisation interference is used in various applications such as:
    • Polarising sunglasses: They reduce glare by selectively blocking horizontally polarised light.
    • Liquid crystal displays (LCDs): The optical properties of liquid crystals change when an electric current is applied, allowing for the control of polarised light to create images.
    • Optical filters: Interference filters selectively transmit or reflect light of certain wavelengths based on the principle of interference.
  1. Interference in Thin Films:
  • Thin films, such as soap bubbles or oil slicks, can create colourful interference patterns due to the reflection and transmission of light at the two interfaces.
  • The colour observed depends on the thickness of the film and the refractive indices of the medium and film.
  • This phenomenon is explained by the mathematical equations for the interference of light waves.
  1. Colours in Thin Films:
  • Thin films produce colours based on the principle of constructive and destructive interference.
  • For a thin film of air, the condition for constructive interference of a certain colour can be determined using the equation: 2nt cosθ = mλ, where n is the refractive index, t is the thickness, θ is the angle of incidence, λ is the wavelength, and m is the order of the fringe.
  1. Newton’s Rings:
  • Newton’s rings is an interference pattern observed when a plano-convex lens is placed on a flat glass plate.
  • The pattern consists of a series of concentric rings with alternating bright and dark regions.
  • It occurs due to the interference of light reflected from the upper and lower surfaces of the air wedge between the lens and the plate.
  1. Michelson Interferometer:
  • The Michelson interferometer is a device used to measure small displacements, determine refractive indices, and detect small changes in wavelength.
  • It consists of a beam splitter, two mirrors, and an observer.
  • By creating interference between two beams of light, information about the path difference and phase difference can be obtained.
  1. Young’s Double-Slit Experiment with Polarised Light:
  • Young’s double-slit experiment can also be performed with polarised light.
  • When polarised light is used, the resulting interference pattern exhibits dark and bright fringes with different polarisation orientations.
  • This experiment helps to understand the effect of polarisation on interference phenomena.

Slide 21: Introduction to Quantum Mechanics

  • Quantum Mechanics is a branch of physics that deals with the behavior of particles at the atomic and subatomic levels.
  • It was developed in the early 20th century to explain phenomena that classical physics couldn’t account for.
  • Quantum Mechanics provides a mathematical framework for describing the probabilistic nature of particles and their wave-particle duality.

Slide 22: Basic Concepts of Quantum Mechanics

  • Wave-particle duality: Particles can exhibit both wave-like and particle-like properties.
  • Superposition: Particles can exist in multiple states simultaneously, with a probability associated with each state.
  • Quantum Uncertainty: The position and momentum of a particle cannot be precisely determined at the same time.
  • Quantum Entanglement: Particles can become entangled, such that the state of one particle is related to the state of another, even if they are physically separated.

Slide 23: Schrödinger Equation

  • The Schrödinger equation is the fundamental equation of quantum mechanics.
  • It describes the behavior of quantum systems, including particles and waves.
  • The equation is a partial differential equation that relates the wave function of a system to its energy.
  • Solving the Schrödinger equation yields the allowed energy levels and wave functions for a given system.

Slide 24: Wave Function and Probability Density

  • The wave function (Ψ) describes the state of a quantum system.
  • It contains information about the position, momentum, and energy of the system.
  • The probability density (|Ψ|^2) is obtained by taking the modulus squared of the wave function.
  • The probability density represents the likelihood of finding a particle in a particular region of space.

Slide 25: Wave Function Normalization

  • The wave function must be normalized, meaning that the integral of the probability density over all space is equal to 1.
  • Normalization ensures that the total probability of finding a particle is 100%.
  • The normalization condition for the wave function is given by: ∫ |Ψ|^2 dV = 1, where dV represents the volume element.

Slide 26: Operators in Quantum Mechanics

  • Operators in quantum mechanics are mathematical quantities that act on the wave function of a system.
  • Operators represent physical observables, such as position, momentum, and energy.
  • The position operator (x) gives the position of a particle in space.
  • The momentum operator (p) gives the momentum of a particle.

Slide 27: Heisenberg Uncertainty Principle

  • The Heisenberg uncertainty principle states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa.
  • Mathematically, it can be expressed as: ΔxΔp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ (h-bar) is the reduced Planck’s constant.

Slide 28: Wave-Particle Duality

  • Wave-particle duality is a fundamental concept in quantum mechanics.
  • It states that particles, such as electrons or photons, can exhibit both wave-like and particle-like behavior.
  • The duality is described by the wave-particle nature of the particle, where its properties can be described by a wave function or a particle with discrete energy levels.

Slide 29: Quantum Tunneling

  • Quantum tunneling is a phenomenon in which a particle passes through a potential energy barrier that would be classically impossible.
  • It occurs due to the wave nature of particles, which allows them to “tunnel” through regions of low potential energy.
  • Quantum tunneling is important in various physical processes, such as radioactive decay and electron transport in quantum devices.

Slide 30: Applications of Quantum Mechanics

  • Quantum mechanics has numerous applications in various fields, including:
    1. Quantum Computing: Using quantum bits (qubits) to perform calculations that are exponentially faster than classical computers.
    2. Quantum Cryptography: Secure communication using the principles of quantum mechanics to ensure the encryption is unbreakable.
    3. Quantum Sensors: High-precision sensors for measuring quantities such as time, position, and magnetic fields.
    4. Quantum Optics: Studying the behavior of light and its interaction with matter at the quantum level.
    5. Quantum Chemistry: Understanding chemical reactions and molecular properties based on quantum mechanical principles.
    6. Quantum Biology: Investigating quantum processes in biological systems, such as photosynthesis and bird navigation.