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 $\lambda<em>m$, then falls. • Increasing $T$ ⇒ $\lambda</em>m$ 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:
  $\Delta \lambda = \lambda' - \lambda = \lambda<em>C(1-\cos\theta),$ where $\lambda</em>C = \dfrac{h}{m_e c} = 0.00243\,\text{nm}$ 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 $K<em>{\max} = h(\nu-\nu</em>0).$
  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: $E<em>{n</em>x n<em>y n</em>z} = \frac{h^{2}}{8m}\left(\frac{n<em>x^{2}}{a^{2}}+\frac{n</em>y^{2}}{b^{2}}+\frac{n_z^{2}}{c^{2}}\right).$

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 $E<em>F$: energy where occupation probability $f</em>{FD}=0.5$ at any T>0; at $T=0$, all states E<EF filled, $E>E</em>F$ 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 $\psi<em>{k}(\mathbf r)=u</em>{k}(\mathbf r) e^{i\mathbf k\cdot\mathbf r}$ with periodic envelope $u<em>{k}(\mathbf r)=u</em>{k}(\mathbf r+\mathbf a).$
• 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 $b\to0, V<em>0\to\infty$ with $V</em>0 b$ 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 $E<em>g$ dictates electrical behaviour: • Conductors: valence & conduction bands overlap ($E</em>g=0$).
  • Semiconductors: small $E<em>g\sim 1{-}2\,\text{eV}$ (Si, Ge, GaAs). • Insulators: large $E</em>g\gtrsim 5\,\text{eV}$ (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.