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Unit 4: Quantum Mechanics

PHY109 — Engineering Physics 9 min read

I. Foundations of Quantum Mechanics

Quantum mechanics, developed chiefly between 1900 and 1927, describes matter and radiation at atomic and subatomic scales. It replaces the deterministic trajectories of classical mechanics with wave functions, quantized observables, and probabilistic predictions.

A. Need for quantum mechanics

Quantum mechanics became necessary because classical physics could not explain several experimentally established microscopic phenomena.

  • Classical assumptions: Classical mechanics assigns a definite position and momentum to a particle at every instant, while classical electrodynamics treats radiation energy as continuously variable.
  • Failure in black-body radiation: Classical theory predicted that emitted energy would diverge at short wavelengths—the ultraviolet catastrophe—whereas measured spectra possess a finite maximum.
  • Atomic stability: An orbiting electron is an accelerating charge and, classically, should radiate continuously, lose energy, and collapse into the nucleus; stable atoms contradict this prediction.
  • Discrete atomic spectra: Excited gases emit sharp spectral lines rather than a continuous spectrum, indicating discrete atomic energies.
  • Wave-particle duality: Photoelectric and Compton effects reveal particle-like photons, while electron diffraction reveals wave-like behavior of matter.
  • Quantum framework:
    • Energy can occur in discrete values called quanta.
    • A system is represented by a wave function.
    • Measurement outcomes are generally probabilistic.
    • Certain pairs of observables cannot simultaneously have arbitrary precision.

II. Black-Body Radiation

A black body is an ideal object that absorbs all incident electromagnetic radiation and, in thermal equilibrium, emits a spectrum determined only by its absolute temperature.

A. Black-body radiation: spectral distribution

The spectral distribution gives the emitted energy per unit wavelength interval as a function of wavelength and temperature.

  • Planck’s hypothesis: Oscillators exchange energy only in discrete packets:
    TEXT
    E_n = n hν,     n = 0, 1, 2, ...

    Here, E_n is the permitted energy, n is a non-negative integer, h is Planck’s constant, and ν is frequency.
  • Planck distribution:
    TEXT
    u(λ,T) = [8πhc/λ^5] / [exp(hc/λk_B T) - 1]

    Here, u(λ,T) is spectral energy density per unit wavelength, λ is wavelength, T is absolute temperature, c is light speed, and k_B is Boltzmann’s constant.
  • Shape of the spectrum: The intensity begins near zero at very short wavelengths, reaches a temperature-dependent maximum, and decreases at long wavelengths.
  • Wien’s displacement law:
    TEXT
    λ_max T = b

    Here, λ_max is the peak wavelength and b = 2.898 × 10^-3 m K. Increasing temperature shifts the peak toward shorter wavelengths.
  • Stefan–Boltzmann law:
    TEXT
    P/A = σT^4

    Here, P/A is total emitted power per unit area and σ is the Stefan–Boltzmann constant.
  • Significance: Planck’s quantization removed the ultraviolet catastrophe and initiated quantum theory.

III. Photoelectric Emission

The photoelectric effect is the emission of electrons from a material when electromagnetic radiation of sufficiently high frequency falls on its surface.

A. Photoelectric effect

The effect demonstrates that radiation transfers energy in localized photons rather than only as a continuous classical wave.

  • Photon energy:
    TEXT
    E = hν = hc/λ

    Here, E is photon energy, ν is radiation frequency, and λ is its wavelength.
  • Einstein’s photoelectric equation:
    TEXT
    hν = φ + K_max
K_max = (1/2)mv_max^2 = eV_s

    Here, φ is the work function, K_max is maximum electron kinetic energy, m is electron mass, v_max is maximum speed, e is elementary charge, and V_s is stopping potential.
  • Threshold frequency:
    TEXT
    ν_0 = φ/h

    No emission occurs for ν < ν_0, irrespective of intensity.
  • Frequency effect: Above threshold, increasing frequency increases maximum kinetic energy and stopping potential.
  • Intensity effect: At fixed frequency above threshold, greater intensity supplies more photons and therefore increases photoelectric current, not K_max.
  • Instantaneous emission: Electrons are emitted without measurable delay because one electron absorbs one photon in a single interaction.

IV. Matter-Wave Hypothesis

Louis de Broglie proposed in 1924 that every moving material particle has an associated wave.

A. Concept of de Broglie matter waves

Matter waves express wave-particle duality by connecting a particle’s momentum with a wavelength.

  • de Broglie relation:
    TEXT
    λ = h/p

    Here, λ is matter wavelength and p is particle momentum.
  • Non-relativistic particle:
    TEXT
    p = mv,     λ = h/(mv)

    Here, m is rest mass and v is particle speed, provided v is much smaller than c.
  • Physical meaning: Matter waves are probability waves associated with a particle; they are not mechanical vibrations of a material medium.
  • Experimental evidence: Davisson and Germer observed electron diffraction from a nickel crystal, verifying the predicted electron wavelength.
  • Scale dependence: Matter wavelengths are appreciable for electrons and neutrons but negligibly small for macroscopic bodies because their momenta are large.

V. Forms of Matter Wavelength

The de Broglie relation can be rewritten using a particle’s kinetic energy, accelerating voltage, or relativistic momentum.

A. Wavelength of matter waves in different forms

Different experimental data lead to equivalent wavelength formulas under their stated conditions.

  • In terms of momentum:
    TEXT
    λ = h/p
  • In terms of non-relativistic kinetic energy:
    TEXT
    K = p^2/(2m),     λ = h/√(2mK)

    Here, K is kinetic energy and m is particle mass.
  • Charged particle accelerated through voltage:
    TEXT
    qV = K,     λ = h/√(2mqV)

    Here, q is charge magnitude and V is accelerating potential.
  • Electron engineering form:
    TEXT
    λ = 12.27/√V Å

    This non-relativistic formula uses V in volts and gives wavelength in ångströms (1 Å = 10^-10 m).
  • Relativistic form:
    TEXT
    λ = h/(γmv),     γ = 1/√(1 - v^2/c^2)

    Here, γ is the Lorentz factor. Relativistic correction becomes important when particle speed is not negligible compared with c.

VI. Propagation of Quantum Waves

A localized quantum particle is represented by a wave packet formed by superposing waves with nearby frequencies and wave numbers.

A. Concept of phase velocity and group velocity (qualitative)

Phase velocity describes the motion of individual wave phases, whereas group velocity describes the motion of the packet envelope.

  1. Phase velocity:

    TEXT
    v_p = ω/k = νλ
    • v_p is phase velocity, ω is angular frequency, and k is angular wave number.
    • Using E = ħω and p = ħk, one obtains v_p = E/p.
    • For a free relativistic particle, phase velocity may exceed c; this does not transmit information.

  2. Group velocity:

    TEXT
    v_g = dω/dk
    • v_g is group velocity and normally represents the speed of the wave packet and associated particle.
    • For a free particle, v_g equals the particle velocity.
    • Dispersion occurs when component waves have different phase velocities, causing the packet to spread.

VII. Limits of Simultaneous Measurement

Werner Heisenberg formulated a fundamental limit on the simultaneous precision of certain conjugate observables in 1927.

A. Heisenberg uncertainty principle

The principle states that a quantum state cannot possess arbitrarily sharp values of both position and momentum along the same direction.

  • Position–momentum relation:
    TEXT
    Δx Δp_x ≥ ħ/2

    Here, Δx and Δp_x are standard deviations of position and momentum, while ħ = h/(2π).
  • Energy–time relation:
    TEXT
    ΔE Δt ≥ ħ/2

    Here, ΔE is energy uncertainty and Δt characterizes the relevant evolution or lifetime.
  • Interpretation: The uncertainty is intrinsic to the quantum state, not merely a defect of measuring instruments.
  • Wave-packet origin: Strong localization requires many wave numbers; because p = ħk, a narrow position distribution produces a broad momentum distribution.
  • Consequence: An electron cannot have both an exact orbital position and exact momentum, invalidating classical atomic trajectories.

VIII. Quantum State Description

The wave function is the mathematical object containing the available information about a quantum system.

A. Wave function and its significance

A wave function ψ(r,t) assigns a complex probability amplitude to position r at time t.

  • Born interpretation:
    TEXT
    Probability density = |ψ(r,t)|^2 = ψ*(r,t)ψ(r,t)

    Here, ψ* is the complex conjugate of ψ.
  • Probability in a volume:
    TEXT
    P = ∫_V |ψ|^2 dτ

    Here, P is the probability of finding the particle in volume V, and dτ is a volume element.
  • Normalization:
    TEXT
    ∫_all space |ψ|^2 dτ = 1

    This expresses certainty that the particle exists somewhere in the allowed region.
  • Acceptability conditions: A physical wave function must be finite, single-valued, normalizable, and generally continuous with a suitably continuous first derivative.
  • Observable meaning: ψ itself is not directly observable; measurable probabilities and expectation values are obtained from it.
  • Superposition: If ψ_1 and ψ_2 are permitted states, then a linear combination can also be a permitted state.

IX. Schrödinger Wave Equations

Schrödinger’s equation governs the evolution of a non-relativistic quantum state under a specified potential energy.

A. Schrodinger time-dependent and time-independent equations

The time-dependent equation is fundamental, while the time-independent equation applies to stationary states in a time-independent potential.

  1. Time-dependent equation:

    TEXT
    iħ ∂ψ/∂t = [-(ħ^2/2m)∇^2 + V]ψ
    • i is the imaginary unit, m is particle mass, ∇² is the Laplacian, and V is potential energy.
    • It determines how an initial wave function evolves with time.

  2. Time-independent equation:

    TEXT
    -(ħ^2/2m)∇^2φ + Vφ = Eφ
    • φ is the spatial eigenfunction and E is an allowed energy eigenvalue.
    • It follows when V is time-independent and:
      TEXT
      ψ(r,t) = φ(r) exp(-iEt/ħ)
    • Stationary probability: For an energy eigenstate, |ψ|² = |φ|² is independent of time.

X. Particle Confined in One Dimension

The infinite one-dimensional box models a particle trapped between perfectly impenetrable walls separated by distance L.

A. Particle in a box

Confinement and boundary conditions produce discrete energies and standing-wave eigenfunctions.

  • Potential model: V(x) = 0 for 0 < x < L, while V(x) is infinite outside; therefore, the particle cannot exist beyond the walls.
  • Boundary conditions:
    TEXT
    ψ(0) = 0,     ψ(L) = 0
  • Normalized eigenfunctions:
    TEXT
    ψ_n(x) = √(2/L) sin(nπx/L),     n = 1, 2, 3, ...

    Here, n is the quantum number and L is box length.
  • Quantized energies:
    TEXT
    E_n = n^2π^2ħ^2/(2mL^2) = n^2h^2/(8mL^2)
  • Ground-state energy: The minimum energy is E_1 = h²/(8mL²), not zero; zero energy would conflict with confinement and the uncertainty principle.
  • Energy spacing:
    TEXT
    E_(n+1) - E_n = (2n + 1)h^2/(8mL^2)

    The levels are discrete and become farther apart as n increases.
  • Probability structure: State n has n - 1 internal nodes, where ψ_n = 0; the probability density is |ψ_n|².
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