Quantum Physics – Comprehensive Lecture Notes

Introduction to Quantum Physics

• Classical mechanics (Newtonian laws + classical electromagnetism) fails in the sub-microscopic (atomic) domain.
• Quantum Mechanics (mathematically formalised 1925–1926) explains:
  • Electronic energies in solids, black-body spectrum, atomic & solid spectra.
  • Magnetism, superconductivity, photoelectric & Compton effects, etc.
• Unit-I roadmap: black-body radiation → Planck’s law → photoelectric effect → Compton effect → de-Broglie hypothesis → Heisenberg uncertainty → Schrödinger equation + applications.

Black-Body Radiation

• Black body: ideal object absorbing 100\% of incident radiation (no reflection/transmission) – appears perfectly black.
• Stars approximate black-body radiators; entire spectrum depends only on temperature T.
• Experimental observations (energy-density vs. wavelength curves):
  • Non-uniform spectral energy distribution.
  • At fixed T, intensity rises with \lambda, peaks at \lambdam, then falls. • Increasing T ⇒ \lambdam shifts lower (Wien’s displacement).
  • Total emitted energy (area under curve) grows with T.

Planck’s Quantum Hypothesis & Radiation Law

• Assumptions:
  1. Cavity walls contain a huge set of electromagnetic oscillators, each of frequency \nu.
  2. Allowed energies: E_n = n h \nu\ \,(n = 0,1,2,\dots).
  3. Emission/absorption in discrete quanta \Delta E = h\nu.
• Average energy per oscillator (derivation via Boltzmann statistics):
  \langle E \rangle = \frac{h\nu}{\exp\left(\dfrac{h\nu}{kT}\right)-1}.
• Number of modes per unit volume in \nu \to \nu+d\nu: \dfrac{8\pi \nu^{2}}{c^{3}}d\nu.
• Planck’s spectral energy-density:
  u(\nu,T)\,d\nu = \frac{8\pi h \nu^{3}}{c^{3}}\frac{1}{\exp\left(\dfrac{h\nu}{kT}\right)-1}\,d\nu.
• In wavelength form:
  u(\lambda,T)\,d\lambda = \frac{8\pi h c}{\lambda^{5}}\frac{1}{\exp\left(\dfrac{hc}{\lambda kT}\right)-1}\,d\lambda.
• Limiting cases reproduced:
  • Wien’s law (short-\lambda, large hc/\lambda kT).
  • Rayleigh–Jeans law (long-\lambda, small hc/\lambda kT).

Compton Effect

• Inelastic scattering of high-energy photons (X/γ) by (free) electrons.
• Shift in wavelength:
  \Delta \lambda = \lambda' - \lambda = \lambdaC(1-\cos\theta), where \lambdaC = \dfrac{h}{m_e c} = 0.00243\,\text{nm} is the Compton wavelength of the electron and \theta the scattering angle.
• Special cases:
  • \theta = 0^\circ → \Delta\lambda = 0 (no shift).
  • \theta = 180^\circ → maximum shift 2\lambda_C.

Photoelectric Effect

• Emission of electrons from a metal when illuminated by light of frequency \nu.
• Observations / laws:
  1. Existence of threshold frequency \nu_0 (cut-off) specific to metal.
  2. Photo-current ∝ light intensity (above \nu_0).
  3. Maximum kinetic energy K{\max} = h(\nu-\nu0).
  4. Emission occurs without time lag.
• Experimental setup: evacuated tube with cathode (illuminated) & anode, bias control, ammeter.

Wave–Particle Duality

• Radiation acts as waves (interference, diffraction, polarisation) & as particles (photons: black-body, photoelectric, Compton).
• Cannot display both characters simultaneously in a single experiment.

de-Broglie Hypothesis (Matter Waves)

• For any particle of momentum p:
  \lambda = \frac{h}{p} = \frac{h}{mv}.
• For kinetic energy E (non-relativistic):
  \lambda = \frac{h}{\sqrt{2mE}}.
• Accelerated electron (potential V):
  \lambda = \frac{12.25}{\sqrt{V}}\,\text{Å} \quad(\text{non-relativistic}).
• Properties:
  • \lambda larger for lighter or slower particles; \lambda\to\infty when v\to0.
  • Wave velocity \omega = c^{2}/v > c (phase velocity; no violation of relativity).
  • No experiment reveals both particle & wave aspects at once.

Davisson–Germer Experiment (Electron Diffraction)

• Electron gun → mono-energetic beam → Ni crystal → movable detector.
• At V = 54 V, intensity peak at 50^{\circ}; using Bragg’s law (d_{\text{Ni}}=0.091\,\text{nm}) gives \lambda = 0.165\,\text{nm}.
• de-Broglie prediction \lambda = 12.25/\sqrt{54} = 0.166\,\text{nm} – excellent agreement; confirms matter waves.

Heisenberg Uncertainty Principle

• Fundamental limit: simultaneous measurement errors obey
  \Delta x\,\Delta p \ge \frac{\hbar}{2}, \qquad \Delta E\,\Delta t \ge \frac{\hbar}{2}.
• Narrow wave packet ⇒ good position, poor momentum; wide packet ⇒ opposite.
• Classical “exact” trajectory impossible at atomic scale.

Schrödinger Wave Equation (Time-Independent)

• Starting from classical wave + de-Broglie relations:
  \nabla^{2}\psi + \frac{2m}{\hbar^{2}}(E-V)\psi = 0.
• In 1-D: \frac{d^{2}\psi}{dx^{2}} + \frac{2m}{\hbar^{2}}(E-V)\psi =0.

Physical Significance of \psi

• Born interpretation: |\psi|^{2} = probability density.
• Normalisation: \int_{-\infty}^{\infty} |\psi|^{2} d\tau = 1.
• Acceptable \psi must be finite, single-valued, continuous with continuous first derivative.

Particle in a 1-D Infinite Potential Box

• Potential: V=0 for 0<x<a, V=\infty elsewhere.
• Boundary conditions \psi(0)=\psi(a)=0.
• Wavefunctions:
  \psi_n(x)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right),\qquad n=1,2,3,\dots
• Quantised energies:
  E_n = \frac{n^{2} h^{2}}{8ma^{2}}.
• Nodes: n+1 per \psi_n; probability densities show standing-wave patterns.
• 3-D box: E{nx ny nz} = \frac{h^{2}}{8m}\left(\frac{nx^{2}}{a^{2}}+\frac{ny^{2}}{b^{2}}+\frac{n_z^{2}}{c^{2}}\right).

Electron Theory of Solids

• Three evolutionary stages:
  1. Classical free-electron theory (Drude-Lorentz, 1900): electrons behave like ideal gas; explains Ohm’s law, some thermal/optical properties; fails for specific heat, superconductivity, etc.
  2. Quantum free-electron theory (Sommerfeld, 1928): includes Pauli & Fermi–Dirac; elastic scattering, constant potential; still cannot classify conductors/semiconductors/insulators.
  3. Zone (Band) theory (Bloch, 1928): electrons in periodic potential; explains band formation, effective mass, etc.

Statistical Distribution Functions

• Maxwell–Boltzmann (MB): distinguishable particles, classical; f_{MB}(E) \propto e^{-E/kT}.
• Bose–Einstein (BE): identical bosons (integer spin), no Pauli exclusion;
  f_{BE}(E)=\frac{1}{\exp\big[(E-\mu)/kT\big]-1}.
• Fermi–Dirac (FD): identical fermions (half-integer spin), obey Pauli;
  f_{FD}(E)=\frac{1}{\exp\big[(E-\mu)/kT\big]+1}.

Fermi Energy & Level

• Fermi level EF: energy where occupation probability f{FD}=0.5 at any T>0; at T=0, all states E<EF filled, E>EF empty.

Periodic Potential, Bloch Theorem & Brillouin Zones

• Real crystal: periodic ion cores ⇒ periodic potential V(\mathbf r)=V(\mathbf r+\mathbf a).
• Bloch theorem: electron eigenfunctions \psi{k}(\mathbf r)=u{k}(\mathbf r) e^{i\mathbf k\cdot\mathbf r} with periodic envelope u{k}(\mathbf r)=u{k}(\mathbf r+\mathbf a).
• Reciprocal-space partitioning yields Brillouin zones; 1st zone spans k\in[-\pi/a,\pi/a].

Kronig–Penney Model (1-D Periodic Square Potential)

• Simplified periodic potential: width a (well), barrier b of height V_0.
• Dispersion relation (after simplifying for b\to0, V0\to\infty with V0 b finite):
  (P/\alpha a)\sin(\alpha a)+\cos(\alpha a)=\cos(ka),
  where \alpha=\sqrt{2mE}/\hbar and P=\frac{mV_0ba}{\hbar^{2}}.
• Allowed energies exist only when LHS within [-1,1] → formation of allowed bands & forbidden gaps.
  • P\to\infty ⇒ discrete atomic-like levels E_n=n^{2}h^{2}/8ma^{2}.$
  • P\to0 ⇒ free-electron parabola E=p^{2}/2m.$

Effective Mass

• From band dispersion E(k):
  m^{*}=\hbar^{2}\left(\frac{d^{2}E}{dk^{2}}\right)^{-1}.
• Near band extrema, E(k)\approx E_0+\frac{\hbar^{2}k^{2}}{2m^{*}}.
• Sign & magnitude vary with curvature; negative curvature ⇒ negative effective mass (interpreted via holes).

Origin of Energy Bands & Classification of Solids

• As N atoms approach, each atomic level splits into N closely-spaced levels → forms bands; energy gaps arise where no permissible states.
• Key bands:
  • Valence band: highest filled (at T=0) – governs bonding & many properties.
  • Conduction band: next higher empty band; carriers here conduct.
• Band gap Eg dictates electrical behaviour: • Conductors: valence & conduction bands overlap (Eg=0).
  • Semiconductors: small Eg\sim 1{-}2\,\text{eV} (Si, Ge, GaAs). • Insulators: large Eg\gtrsim 5\,\text{eV} (glass, rubber).

Symmetry Operations in Crystals

• Centre of symmetry (inversion centre).
• Plane (mirror) of symmetry.
• Axis of symmetry: n-fold rotation giving identical configuration (n = 2,3,4,6).

Formulae & Quick Reference (as in transcript)

1. Photon wavelength: \lambda = \dfrac{hc}{E}.
2. General de-Broglie \lambda = h/p.
3. \lambda = h/\sqrt{2mE}.
4. \lambda = h/\sqrt{2m kT}.
5. Electron accelerated through V: \lambda = 12.25/\sqrt{V}\,\text{Å}.
6. Particle in 1-D box energies: E_n=n^{2}h^{2}/8ma^{2}.
7. Schrödinger T-independent: -\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi=E\psi.
8. Uncertainty (x–p): \Delta x\,\Delta p\ge \hbar/2.
9. Compton shift: \Delta\lambda=\lambda_C(1-\cos\theta).

Typical Exam Questions (from transcript)

• Short answers: black body definition, photoelectric effect, Compton shift, Planck postulates, wave vs. particle, properties of matter waves, uncertainty principle, significance of \psi, drawbacks of Drude model, electrons in periodic potential, energy-band origin, crystal symmetries.
• Essays: Derive Planck law; Compton shift; Davisson–Germer verification; derive Schrödinger equation; particle-in-box solution; Kronig–Penney → bands; effective mass derivation; compare conductors, semiconductors, insulators.
• Multiple-choice examples include Wien region, uncertainty inequalities, identification of matter waves, wave-function meaning, experimental confirmations, etc.