QUANTUM PHYSICS AND COMPUTATION FOR COMPUTER SCIENCE ENGINEERING STREAM

QUANTUM PHYSICS AND COMPUTATION FOR COMPUTER SCIENCE ENGINEERING STREAM

MODULE 1: QUANTUM MECHANICS

• Imagination and Knowledge

  • Quote by Sir Albert Einstein: "Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world."

DE-BROGLIE HYPOTHESIS – DERIVATION BY ANALOGY

• Origin of Energy Quantization

  • The dual nature (particle and wave) observed in nature.

  • Example: X-rays behavior observed by Compton during scattering with electrons.

    • X-rays behave as particles (due to elastic collisions)

    • X-rays diffract in crystals behaving as waves.

  • Louis de Broglie Hypothesis (1924):

    • Dual behavior suggests that particles can exhibit wave properties.

    • Termed as "matter waves."

  • Equations:

    • Energy of a photon: $E = h<br>u$ (wave nature)

    • Mass-energy relation: $E = mc^2$ (particle nature)

  • By comparing the above equations: $mc^2 = h u ext{ where } c = u imes ext{ wavelength } ext{(} u = rac{c}{ ext{λ}} ext{)}$

    • Rearranging gives the relation between wavelength and momentum:

    • Matter Waves Relationship:
      $ext{λ} = rac{h}{p}$

DE BROGLIE WAVELENGTH OF AN ACCELERATED ELECTRON

• Formula:

  • De Broglie wavelength for matter waves: $ext{λ} = rac{h}{p}$

  • High electron velocity corresponds to smaller de Broglie wavelength.

  • When an electron is accelerated through a potential difference (V):

    • Work done on the electron: $ext{Work} = eV$

    • This converts to kinetic energy:
      $rac{1}{2} mv^2 = eV$

  • Rearranging results in:
    $mv = \bigg(2meV\bigg)^{1/2}$

  • Substituting in the de Broglie wavelength gives:
    $ext{λ} = rac{h}{(2meV)^{1/2}}$

  • This wavelength can also be expressed in Ångströms (Å).

RELATION BETWEEN KINETIC ENERGY AND DE BROGLIE WAVELENGTH

• Kinetic Energy (E):

  • Total energy relation: $E = eV$

• Phase Velocity:

  • Concept: If a wave is traveling, the velocity at which the phase of the wave propagates is called phase velocity.

  • Written as:
    $v_{ ext{phase}} = rac{ ext{ω}}{k}$

  • For a wave described by:
    $Y = A ext{sin} ( ext{ωt} - kx)$

GROUP AND PHASE VELOCITY

• Group Velocity:

  • Defined when a wave packet or group consists of numerous component waves traveling at slightly different velocities.

  • Group velocity formula: $v_g = rac{d ext{ω}}{dk}$

  • Relationship to wave velocity:

    1. $v<em>g ext{ and } v</em>p$ (where v_p = phase velocity)

    2. $v<em>g = v</em>p + k$

• Understanding the group velocity and phase velocity in relation to potential waves and their properties.

HEISENBERG’S UNCERTAINTY PRINCIPLE

• Concept:

  • A particle can be represented as a packet or group of waves.

  • Cannot accurately find a particle's position and momentum simultaneously.

• Statement:

  • Uncertainty Relation:
    $ext{Δx} ext{Δp}_x ext{ ≥ } rac{h}{4 ext{π}}$

  • Variants of this concept apply to various physical variables.

PHYSICAL SIGNIFICANCE

• Understanding the probabilistic nature of quantum mechanics in determining a particle's trajectory and location.

APPLICATIONS OF THE HEISENBERG UNCERTAINTY PRINCIPLE

NON-EXISTENCE OF ELECTRONS IN NUCLEUS

• Using uncertainty to determine if an electron can exist in a nucleus of size $ext{Δx} ext{ ≤ } 10^{-14} m$.

• Relations from energy momentum equations and calculated bounds on electron energy.

WAVE FUNCTION (ψ)

• Definition: The wave function represents a particle's quantum state and its probability density.

• Max Born Interpretation:

  • Probability of locating a particle within a given space.

  • $|ψ|^2$ provides the probability density.

  • Valid interpretation only works under certain conditions, as amplitude can vary in sign.

TIME INDEPENDENT ONE DIMENSIONAL SCHRÖDINGER WAVE EQUATION

• Key equation for de Broglie waves and kinetic energies is introduced, alongside its derivation.

EIGEN VALUE & EIGEN FUNCTION

• Wave functions become eigen functions under certain operations, leading to eigen values expected from Schrödinger's equations based on particle confinement.

• Discuss potential wells and derived energies from wave functions laid out in 1-D conditions.

ENERGY EIGEN VALUES

• Trapped particles in potential wells are discussed, including calculated energy states and their implications for probability densities.

  • Systems yielding quantized energy levels in bounded conditions are explored.

NUMERICAL EXAMPLES

• Examples and calculations of energy in various physical scenarios, including potential wells, electron transitions, photon calculations, particle confinement, and velocity relations in quantum mechanics.