Wave–Particle Duality, Black-Body Radiation & Quantum Foundations

Quantum Mechanics & Quantization

• Name originates from Latin “quantum / quanta” = “how much; discrete amount”.

• Key postulate: many physical properties (energy, momentum, angular momentum, charge, …) change only in discrete packets rather than continuously.

• Quantum (singular) / Quanta (plural):
  • Minimum, indivisible unit of a physical property involved in an interaction.
  • Smallest possible unit of energy; energy stored in discrete packets.
  • In Planck’s discovery the packet energy is $E=h\nu$.

• Quantization (general definition): constraining a continuous set of values to a discrete set → transition from classical physics to quantum mechanics.

Black-Body Radiation

Definitions & Model System

• Black body: ideal object that absorbs $100\%$ of incident radiation (no reflection) → appears perfectly black at room T.

• White body: reflects $100\%$ of incident radiation (no absorption).

• Practical realization: a small hole in a cavity; multiple internal reflections → incident radiation is totally absorbed; the hole itself appears black.

Thermal Equilibrium & Emission Characteristics

• Black-body reaches equilibrium when absorbed power = radiated power.

• Emits a continuous distribution of wavelengths; total spectrum depends only on temperature, not material.

• All objects above absolute zero emit radiation; examples:
  • $300\,\text{K}$ → infrared only (object looks black to eye).
  • $600!\text{–}!700\,\text{K}$ → dull red glow.
  • Very high T → white-hot → blue-white region.

• Peak wavelength shifts with temperature (Wien’s displacement): \lambda{\text{max}}\,T \approx 2.898\times10^{-3}\;\text{m·K}. • Example: $T=1250\,\text{K}\Rightarrow\lambda</em>{\text{max}}\approx2.5\,\mu m$; $T=2000\,\text{K}\Rightarrow\lambda_{\text{max}}\approx1.5\,\mu m$.

• Graphical description: family of black-body curves (intensity vs. $\lambda$); each curve peaks, then falls to 0 at short & long wavelengths.

• Sun behaves almost like a perfect black body at surface temperature $5778\,\text{K}$ → intense visible–UV emission.

• Colour thermometry: lava colour → temperature scale (1000–10 000 K chart).

Classical Attempts & Ultraviolet Catastrophe

• 19th-century challenge: predict $u(\lambda,T)$ (spectral energy density).

• Wien’s law: good at short wavelengths, fails at long.

• Rayleigh-Jeans law: assumes cavity EM standing waves → $u_{RJ}(\lambda,T)=\dfrac{8\pi kT}{\lambda^{4}}$
  • Fits long-wavelength data, diverges $\rightarrow\infty$ as $\lambda\to0$ → “ultraviolet catastrophe”.

• Experimental data show finite peak; classical theory gives no peak.

Planck’s Resolution (1900)

• Assumptions:

  1. Wall atoms = harmonic oscillators of all possible $\nu$.

  2. Oscillator emits/absorbs EM radiation at its own $\nu$.

  3. Energy exchange occurs in integral packets $E_n=n h\nu$ ( $n=0,1,2,\dots$ ).

• Planck spectrum (frequency form):
  $u(\nu,T)=\frac{8\pi h\nu^{3}}{c^{3}}\;\frac{1}{\exp\left(\dfrac{h\nu}{kT}\right)-1}$.

• Wavelength form (page 27):
  $u(\lambda,T)\,d\lambda=\frac{8\pi h c}{\lambda^{5}}\;\frac{1}{\exp\left(\dfrac{hc}{\lambda kT}\right)-1}\,d\lambda$.

• Correctly fits entire spectrum; ended ultraviolet catastrophe; birth of quantum theory.

Applications

• Astronomy: star temperatures via peak $\lambda_{\text{max}}$.

• Calibration standards: IR thermometers, pyrometers.

• Thermal imaging: cameras map $T$ from IR intensity.

• Material science: deduce composition/structure from spectral emissivity.

• Technology: incandescent bulbs, electric heaters, furnace design; efficiency optimisation relies on black-body theory.

• Historical importance: led directly to quantum mechanics.

Failures of Classical Physics (Motivation for Quantum Theory)

1. Black-body spectrum.

2. Discrete line spectra of atoms.

3. Temperature-dependent specific heats of solids & gases.

4. Atomic stability.

5. Photoelectric, Compton, Zeeman, Raman effects, etc.

Wave–Particle Duality of Radiation

• Wave evidence: diffraction, interference, polarization.

• Particle evidence: photoelectric effect, Compton scattering.

• Conclusion: EM radiation shows dual character.

Photoelectric Effect

Phenomenon & History

• Discovered (observation) by Edmund Becquerel 1839; explained by Einstein 1905 (Nobel Prize).

• Light incident on metal surface ejects electrons (photoelectrons) if frequency exceeds threshold.

Einstein’s Photon Picture

• Each photon energy $E_{\gamma}=h\nu$ transfers to one electron.

• Energy balance: $h\nu = \phi + K<em>{\text{max}}$, where • $\phi$ = work function (minimum energy to liberate electron). • $K</em>{\text{max}}=\dfrac{1}{2}m v<em>{\text{max}}^{2}$ measured via stopping potential $V</em>0$ : $K<em>{\text{max}}=e V</em>0$.

Experimental Setup

• Vacuum tube with photosensitive cathode & anode, adjustable retarding potential $V$, ammeter, rheostat.

Laws of Photoelectric Emission (Lenard & Millikan)

1. For given metal & \nu>\nu_0 (threshold), photocurrent $\propto$ incident intensity (saturation current directly proportional).

2. Existence of threshold frequency $\nu<em>0$ (or threshold wavelength $\lambda</em>0$) below which emission impossible, irrespective of intensity.

3. For \nu>\nu0, stopping potential $V</em>0$ (hence $K<em>{\text{max}}$) varies linearly with $\nu$; independent of intensity. $K</em>{\text{max}} = h\nu - h\nu_0$.

4. Emission is instantaneous (time lag < $10^{-9}\,\text{s}$).

Key Observations

• Increasing intensity increases number of photons → more electrons, but $K_{\text{max}}$ unchanged.

• Increasing frequency (above $\nu<em>0$) keeps electron count same (if intensity constant) but grows $K</em>{\text{max}}$.

• Example (Potassium): $\phi\approx2.0\,\text{eV}$; red light $700\,\text{nm}\,(1.77\,\text{eV})$ fails; green or violet succeeds.

Applications

• Solar cells, photomultipliers, light meters, automatic doors, etc.

Matter Waves (de Broglie Hypothesis, 1924)

• Symmetry argument: if light (waves) shows particle nature, matter (particles) should show wave nature.

• A moving particle of momentum $p$ has wavelength (pilot wave)
  $\lambda = \frac{h}{p}$.

Properties of Matter Waves

1. Lighter particle → larger $\lambda$.

2. Slower particle → larger $\lambda$.

3. Not electromagnetic; produced by particle motion (none for rest).

4. Independent of particle charge; depend on momentum.

5. Phase/group velocities can exceed $c$; velocity not constant.

6. “Pilot” or guiding waves; represent probability, not physical oscillation.

Heisenberg Uncertainty Principle (1927)

• Statement: certain conjugate pairs cannot be simultaneously known with arbitrary precision.
  $\Delta x\,\Delta p \ge \frac{\hbar}{2}$,
  $\Delta E\,\Delta t \ge \frac{\hbar}{2}$.

• Physical view: particle described by wave packet; narrow position → broad range of wavelengths (momenta), and vice-versa.

• Consequence: cannot build experiment that shows both exact wave & particle aspects simultaneously.

Schrödinger Wave Mechanics

Time-Independent Equation (1-D)

$\frac{\mathrm{d}^{2}\psi}{\mathrm{d}x^{2}} + \frac{2m}{\hbar^{2}}\left(E-V\right)\psi = 0$.

• Provides mathematical description of particle wave nature.

Wave Function $\psi(x,y,z,t)$

• Complex: $\psi = a+ib$ (real functions $a,b$).

• Probability density: $P=|\psi|^{2}=\psi\,\psi^{*}=a^{2}+b^{2}$.

• Probability of finding particle in volume element $dv$: $P\,dv=|\psi|^{2}dv$.

• Normalisation (3-D): $\int_{-\infty}^{+\infty}|\psi|^{2}\,dv = 1$.

• If not satisfied, multiply by normalisation factor $N$: $\psi_{\text{norm}}=N\psi$ with $N=\left(\int|\psi|^{2}dv\right)^{-1/2}$.

• Orthogonality of different states $i \ne j$:
  $\int \psi<em>i^{*}\psi</em>j\,dv = 0$.

• Orthonormal set combines normalisation & orthogonality:
  $\int \psi<em>i^{*}\psi</em>j\,dv = \delta_{ij}$.

• Wave function must be single-valued, finite, continuous (except where potential has infinite discontinuity); tends to 0 at $|x|\to\infty$.

• Degeneracy: different $\psi<em>i,\psi</em>j$ with same energy $E$.

Stern–Gerlach Experiment (1922)

Goal & Concept

• Demonstrate spatial quantization of angular momentum; reveal electron spin $s=\tfrac{1}{2}$ with two projections $m_s=\pm\tfrac{1}{2}$.

Experimental Details

1. Source: silver atoms (heated to $1000\,^{\circ}\text{C}$) produce neutral mono-atomic beam.
   • Choice just one unpaired $5s^{1}$ electron → zero orbital $L=0$ → magnetic moment solely from spin.
   • Chemically inert, no electric-field interaction.

2. Collimation & Slit: narrow beam enters magnet.

3. Magnet: specially shaped poles create inhomogeneous field (field gradient along $z$) → force $\mathbf{F}=\nabla(\mathbf{\mu}\cdot\mathbf{B})$.

4. Detection: particles hit screen/plate; observed two discrete spots, not continuous smear.

Observations & Surprises

• Beam split into exactly two components (spin-up / spin-down).

• Confirms quantized magnetic moment orientations; contradicts classical continuous prediction.

• Path pattern shows “mouth-like” shape due to off-axis trajectories.

Conclusions & Significance

• First direct evidence of intrinsic spin & space quantization.

• Established two-state nature of electron spin.

• Provided foundation for quantum mechanics and later technologies (NMR, MRI, spintronics).

Connections & Broader Implications

• Black-body problem → Planck quantum postulate → foundation of energy quantization.

• Einstein applied quantization to photons (photoelectric effect); photons verified particle aspect of light.

• de Broglie extended duality to matter → wave mechanics (Schrödinger) & probabilistic interpretation.

• Heisenberg formalised uncertainty, setting limits on simultaneous measurements.

• Stern–Gerlach supplied experimental confirmation of quantized angular momentum (spin), enriching quantum theory’s completeness.

Key Constants & Formulae to Memorise

• Planck constant: h = 6.626\times10^{-34}\,\text{J·s}, $\hbar = h/2\pi$.

• Boltzmann constant: k = 1.381\times10^{-23}\,\text{J·K}^{-1}.

• Wien displacement: \lambda_{\text{max}}T = 2.898\times10^{-3}\,\text{m·K}.

• Rayleigh-Jeans, Wien, Planck distributions (see above).

• Photon energy: $E=h\nu = \dfrac{hc}{\lambda}$.

• de Broglie wavelength: $\lambda = \dfrac{h}{p}=\dfrac{h}{mv}$.

• Uncertainty: $\Delta x\,\Delta p \ge \dfrac{\hbar}{2}$, $\Delta E\,\Delta t \ge \dfrac{\hbar}{2}$.

• Photoelectric: $K<em>{\text{max}} = h\nu - \phi = eV</em>0$.

• Planck spectrum (wavelength): $u(\lambda,T)=\dfrac{8\pi h c}{\lambda^{5}}\left[\exp\left(\dfrac{hc}{\lambda kT}\right)-1\right]^{-1}$.

These bullet-point notes cover every major & minor point, definitions, formulas, historical context, experimental details, examples, and implications required for a comprehensive study of Wave–Particle Duality and the foundational quantum-mechanical phenomena described in the transcript.