Quantum mechanics basics

In short

Quantum mechanics is the physics of matter and radiation at atomic scales, where energy comes in discrete quanta, particles show wave behaviour, and the state of a system is a complex wave function whose squared magnitude gives probability. It replaced classical mechanics for electrons, photons and nuclei after blackbody radiation, the photoelectric effect and electron diffraction could not be explained classically. For a B.Sc Physics student it is the core of the Modern Physics paper and carries the largest single block of marks in GATE Physics, JAM, NET/JRF and university semester exams, usually as Schrodinger-equation problems, uncertainty estimates and operator algebra. The same ideas underpin Indian work in semiconductors, lasers, NMR, quantum computing (the National Quantum Mission, approved 2023) and instrumentation at BARC, ISRO, DRDO, TIFR, IISc, RRCAT and the IITs.

Explained

Why classical physics failed — the quantum hypothesis

• Blackbody radiation: classical Rayleigh–Jeans law, u(ν) ∝ ν²T, diverges at high frequency (the "ultraviolet catastrophe"); Wien's law fits only short wavelengths.
• Planck (1900): oscillators in cavity walls exchange energy only in packets E = nhν. His law fits the whole spectrum and reduces to Rayleigh–Jeans at low ν and Wien at high ν (check both limits — a standard exam step).
• Wien displacement law λ_max T = b (b ≈ 2.898 × 10⁻³ m·K) and Stefan–Boltzmann E = σT⁴ follow from Planck's law by differentiation and integration respectively.
• Photoelectric effect (Einstein, 1905): light of frequency ν ejects electrons with K_max = hν − φ, instantaneously, with a threshold frequency; intensity changes the *number* of electrons, not their energy. Classical waves predict none of this.
• Compton effect (1923): X-rays scattered by free electrons shift in wavelength, Δλ = (h/m₀c)(1 − cos θ). Proves the photon carries momentum p = h/λ and that the collision is relativistic two-body kinematics.
• Pair production/annihilation, Raman effect (C. V. Raman, 1928, Nobel 1930) — further direct evidence of photons; the Raman effect is Indian-origin and frequently asked.
• Franck–Hertz experiment: discrete energy loss of electrons in mercury vapour confirms quantised atomic energy levels.

Wave–particle duality and matter waves

• de Broglie (1924): every particle of momentum p has wavelength λ = h/p. Phase velocity v_p = ω/k = c²/v (exceeds c, carries no information); group velocity v_g = dω/dk = v, the particle speed.
• Davisson–Germer (1927): electrons of ~54 eV diffract off a nickel crystal with a peak at 50°, matching Bragg's law with λ from de Broglie — direct proof of matter waves. G. P. Thomson showed the same with thin-film transmission.
• A wave packet built from a spread of k values is localised in space; the spread Δk and width Δx are inversely related — the Fourier-transform origin of uncertainty.
• Duality is complementary: in a given experiment you see wave behaviour or particle behaviour, never both at once (two-slit experiment with which-path detection destroys the fringes).

Uncertainty principle

• It is not an instrumental limitation but a property of conjugate variables; it follows from the commutator [x, p] = iℏ via the generalised relation ΔA·ΔB ≥ ½|⟨[A,B]⟩|.
• Angular form: ΔL_z·Δφ ≥ ℏ/2. Energy–time uncertainty gives the natural linewidth of spectral lines and the width Γ of unstable states, with lifetime τ ≈ ℏ/Γ.
• Standard consequences examined: non-existence of electrons inside the nucleus, ground-state (zero-point) energy of the oscillator and the box, finite size of the atom, and why an electron cannot sit on the proton.

The wave function and its interpretation

• State of a system ↔ complex wave function ψ(x, t). Born rule: |ψ|² dx is the probability of finding the particle in dx; |ψ|² is a probability density (units m⁻¹ in 1-D, m⁻³ in 3-D).
• Normalisation: ∫|ψ|² dx = 1 over all space. A physically acceptable ψ must be single-valued, finite, continuous, with continuous dψ/dx (except where V is infinite), and square-integrable.
• Probability current j = (ℏ/2mi)(ψ*∂ψ/∂x − ψ ∂ψ*/∂x) satisfies ∂|ψ|²/∂t + ∂j/∂x = 0 — conservation of probability.
• Superposition: if ψ₁ and ψ₂ are solutions, so is c₁ψ₁ + c₂ψ₂; expansion in eigenfunctions ψ = Σ cₙ uₙ gives |cₙ|² as the probability of measuring eigenvalue aₙ. Measurement collapses ψ to that eigenfunction.
• Stationary state: ψ(x,t) = u(x) e^−iEt/ℏ; |ψ|² is time-independent, so all probabilities are constant in time.

Operators, eigenvalues and expectation values

• Observables are Hermitian operators: position x̂ = x, momentum p̂ = −iℏ ∂/∂x, kinetic energy −(ℏ²/2m)∇², Hamiltonian Ĥ = −(ℏ²/2m)∇² + V, angular momentum L̂ = r̂ × p̂.
• Hermitian ⇒ real eigenvalues and orthogonal eigenfunctions (∫uₘ*uₙ dx = δₘₙ) — the basis of all expansion methods.
• Expectation value ⟨A⟩ = ∫ψ*Âψ dx for normalised ψ; this is the quantity compared with a laboratory average over many identical measurements.
• Commutators: [x, p] = iℏ; [L_x, L_y] = iℏL_z; [L², L_z] = 0. Commuting operators share eigenfunctions and are simultaneously measurable; non-commuting ones obey an uncertainty relation.
• Ehrenfest theorem: d⟨x⟩/dt = ⟨p⟩/m, d⟨p⟩/dt = −⟨dV/dx⟩ — classical mechanics emerges for expectation values (correspondence principle).
• Degeneracy: different eigenfunctions with the same eigenvalue; a 3-D cubical box has levels of degeneracy 1, 3, 3, 3, 1, 6 ... for (111), (211)-type, etc.

Schrodinger equation and standard solved problems

• Time-dependent: iℏ ∂ψ/∂t = Ĥψ. Time-independent (V independent of t): −(ℏ²/2m)d²u/dx² + Vu = Eu. Method: write the equation region by region, solve, then apply continuity of u and u′ at the boundaries plus normalisation.
• Infinite square well (particle in a box), width L: u_n = √(2/L) sin(nπx/L), E_n = n²π²ℏ²/(2mL²) = n²h²/(8mL²), n = 1, 2, 3 ... Energy is quantised because of boundary conditions; ground state energy ≠ 0 (uncertainty principle). Nodes: n − 1 inside the well.
• Finite well: finitely many bound states; the wave function leaks exponentially into the classically forbidden region; energies are lower than the infinite-well values.
• Step potential (E < V₀): complete reflection (R = 1) but non-zero penetration depth; (E > V₀): partial reflection even though the particle has enough energy — a purely wave effect.
• Rectangular barrier — tunnelling: for E < V₀, transmission T ≈ 16E(V₀−E)/V₀² · e^−2αa with α = √(2m(V₀−E))/ℏ. Explains alpha decay (Gamow), field emission, the tunnel diode, STM and Josephson junctions.
• Harmonic oscillator: E_n = (n + ½)ℏω, n = 0, 1, 2 ... Equally spaced levels, zero-point energy ½ℏω, Hermite-polynomial × Gaussian eigenfunctions, selection rule Δn = ±1; the ground state is the minimum-uncertainty state.
• Free particle: continuous spectrum, plane waves e^ikx, not normalisable in the usual sense (box or delta normalisation used).
• Hydrogen atom (outline only — detail belongs to atomic physics): separation in spherical polar coordinates gives E_n = −13.6/n² eV, quantum numbers n, l, m with l ≤ n − 1, |m| ≤ l.
• Old quantum theory, for contrast: Bohr quantisation mvr = nℏ and the Wilson–Sommerfeld rule ∮p dq = nh are pre-Schrodinger and fail for multi-electron atoms; the correspondence principle requires quantum results to go classical for large n.

Spin, identical particles and formalism

• Spin is an intrinsic angular momentum with s = ½ for electrons, eigenvalues ±ℏ/2 for S_z; evidence from the Stern–Gerlach experiment and the anomalous Zeeman effect. Pauli matrices represent spin-½ operators.
• Angular momentum quantisation: L² eigenvalue l(l+1)ℏ², L_z eigenvalue mℏ; spherical harmonics Y_lm are the eigenfunctions.
• Identical particles: total ψ is symmetric for bosons (integer spin, Bose–Einstein statistics) and antisymmetric for fermions (half-integer spin, Fermi–Dirac statistics); antisymmetry gives the Pauli exclusion principle.
• Dirac notation: state |ψ⟩, inner product ⟨φ|ψ⟩, operator matrix element ⟨φ|Â|ψ⟩; expansion |ψ⟩ = Σ cₙ|n⟩ with cₙ = ⟨n|ψ⟩. The state lives in a Hilbert space; a measurement outcome is an eigenvalue of the corresponding Hermitian operator.
• Approximation methods (where the syllabus includes them): time-independent perturbation theory E_n ≈ E_n⁰ + ⟨n|H′|n⟩, the variational method (any trial function gives an upper bound to the ground-state energy), and WKB for slowly varying potentials.
• Interpretational points asked in interviews: the Copenhagen interpretation, EPR argument and Bell inequalities, entanglement, and decoherence — quantum mechanics is statistically predictive, not deterministic for individual events.

Key points

• Planck quantum - energy exchanged only as E = hν; h = 6.626 × 10⁻³⁴ J·s, ℏ = h/2π = 1.055 × 10⁻³⁴ J·s.
• Photoelectric equation - K_max = hν − φ; threshold ν₀ = φ/h; stopping potential V₀ = (hν − φ)/e.
• Compton shift - Δλ = (h/m₀c)(1 − cos θ); Compton wavelength of the electron h/m₀c = 2.426 pm; maximum shift at θ = 180°.
• de Broglie wavelength - λ = h/p; for an electron accelerated through V volts, λ ≈ 1.226/√V nm (non-relativistic).
• Uncertainty principle - Δx·Δp ≥ ℏ/2; conjugate variables cannot be sharp together; source of zero-point energy.
• Born rule - |ψ|² is probability density; ψ itself has no direct physical meaning and may be complex.
• Admissible wave function - finite, single-valued, continuous ψ and ψ′, square-integrable.
• Stationary state - ψ = u(x)e^−iEt/ℏ; probability density independent of time.
• Hermitian operator - real eigenvalues, orthogonal eigenfunctions; every observable is represented by one.
• Commutator - [x, p] = iℏ; commuting observables are simultaneously measurable, non-commuting are not.
• Particle in a box - E_n = n²h²/(8mL²); E ∝ 1/L², so confinement raises energy (quantum dots).
• Tunnelling - finite transmission for E < V₀, exponentially sensitive to barrier width and √(mass × barrier height).
• Harmonic oscillator - E_n = (n + ½)ℏω, equal spacing ℏω, zero-point energy ½ℏω.
• Angular momentum - L² → l(l+1)ℏ², L_z → mℏ; spin ½ for electrons, confirmed by Stern–Gerlach.
• Pauli exclusion - antisymmetric total wave function for fermions; no two electrons share all quantum numbers.
• Correspondence principle - quantum results reproduce classical ones for large quantum numbers or ℏ → 0 (Ehrenfest theorem).
• Expectation value - ⟨A⟩ = ∫ψ*Âψ dx; matches the average of many measurements, not a single outcome.
• Degeneracy - distinct states of equal energy; broken by a perturbation such as a magnetic field.

Formulas and examples

Example 1 — Photoelectric effect. Light of λ = 400 nm falls on a sodium surface, φ = 2.28 eV. Find K_max and the stopping potential.

• E = 1240/400 = 3.10 eV (using hc = 1240 eV·nm).
• K_max = 3.10 − 2.28 = 0.82 eV, so V₀ = 0.82 V.
• Threshold wavelength λ₀ = 1240/2.28 ≈ 544 nm — green light works, red does not.

Example 2 — de Broglie wavelength. (a) An electron accelerated through 100 V. (b) A 60 kg student walking at 1.5 m/s.

• (a) λ = 1.226/√100 = 0.1226 nm — comparable to crystal plane spacing, hence electron diffraction and the electron microscope.
• (b) p = 90 kg·m/s, λ = 6.626 × 10⁻³⁴/90 ≈ 7.4 × 10⁻³⁶ m — ~10²⁰ times smaller than a nucleus, so no observable wave behaviour.

Example 3 — Particle in a box. An electron confined in a 1-D well of width L = 1 nm. Find E₁ and the 2 → 1 transition wavelength.

• E₁ = h²/(8mL²) = (6.626 × 10⁻³⁴)²/(8 × 9.11 × 10⁻³¹ × (1 × 10⁻⁹)²) = 4.39 × 10⁻⁶⁷/7.29 × 10⁻⁴⁸ ≈ 6.02 × 10⁻²⁰ J ≈ 0.376 eV.
• E₂ = 4E₁ = 1.504 eV; ΔE = 1.128 eV → λ = 1240/1.128 ≈ 1,099 nm (near infrared).
• Shrinking the well to 0.5 nm multiplies every level by 4 — the design principle of quantum dots.

Example 4 — Uncertainty principle. Estimate the minimum kinetic energy of an electron confined to a nucleus, Δx ≈ 5 × 10⁻¹⁵ m.

• Δp ≥ ℏ/(2Δx) = 1.055 × 10⁻³⁴/(1 × 10⁻¹⁴) ≈ 1.055 × 10⁻²⁰ kg·m/s.
• pc ≈ 1.055 × 10⁻²⁰ × 3 × 10⁸ = 3.17 × 10⁻¹² J ≈ 19.8 MeV, far above the few-MeV binding energy per nucleon — relativistic treatment needed, and the conclusion stands: electrons do not pre-exist inside the nucleus; beta particles are created at decay.

Example 5 — Compton scattering. X-rays of λ = 0.0700 nm scatter at θ = 90°.

• Δλ = 2.426 pm × (1 − 0) = 2.426 pm = 0.002426 nm; λ′ = 0.07243 nm.
• Incident photon energy = 1240/0.0700 = 17.71 keV; scattered = 1240/0.07243 = 17.12 keV.
• Electron recoil kinetic energy = 0.59 keV (about 3.3% of the incident energy).

Where it is used

• Semiconductors and electronics: band structure, tunnel and Zener diodes, MOSFET gate leakage, quantum wells in LEDs and laser diodes — relevant to the India Semiconductor Mission fabs and to design roles at SCL Mohali, Intel, AMD, Micron and Applied Materials India.
• Lasers and photonics: population inversion, selection rules and stimulated emission — RRCAT Indore, LASTEC/DRDO, fibre-optic and medical-laser industry.
• Nuclear and space sector: tunnelling in alpha decay and fusion, reactor physics and radiation detectors at BARC, IGCAR, NPCIL; detector and payload physics at ISRO (AstroSat X-ray instruments).
• Analytical instrumentation and chemistry: NMR, ESR, Raman and UV-visible spectroscopy, electron microscopy (TEM/SEM/STM) used across CSIR labs, pharma QC and materials testing.
• Quantum technologies: the National Quantum Mission (approved by the Union Cabinet in 2023, roughly ₹6,000 crore to 2031) funds hubs for quantum computing, communication, sensing and materials at IISc, IITs, TIFR, C-DAC and DRDO's quantum laboratory; QKD testbeds have been demonstrated by ISRO and DRDO.
• Job roles and exams: scientist/engineer entry through GATE Physics, JEST, TIFR GS, JAM and the BARC OCES/DGFS examination; research positions via CSIR-UGC NET/JRF; teaching through UGC NET.

Interview questions

• Q: What does the wave function physically represent? A: Nothing directly measurable; |ψ|² is the probability density of finding the particle, and ψ gives the amplitudes whose squared moduli are measurement probabilities.
• Q: Why can't an electron have zero energy in a box? A: Zero energy means zero momentum and so infinite position uncertainty, which contradicts confinement in width L; the lowest allowed state is E₁ = h²/8mL².
• Q: Is the uncertainty principle due to imperfect instruments? A: No. It is a property of conjugate observables arising from [x, p] = iℏ, and holds for ideal apparatus.
• Q: Why must operators representing observables be Hermitian? A: Measured values are real; Hermiticity guarantees real eigenvalues and a complete orthogonal set of eigenfunctions.
• Q: How does the photoelectric effect disprove the classical wave picture? A: Classically, energy accumulates with intensity and any frequency should eventually eject electrons; experiment shows a sharp threshold frequency, instantaneous emission, and K_max depending only on frequency.
• Q: What experiment proves that particles behave as waves? A: The Davisson–Germer experiment: ~54 eV electrons diffract off a nickel crystal with a maximum at 50°, matching λ = h/p.
• Q: What is quantum tunnelling and where is it used? A: Transmission through a barrier with E < V₀ because ψ decays exponentially rather than vanishing; used in tunnel diodes, flash-memory writing, STM and alpha decay.
• Q: Why do we see no quantum effects in daily life? A: The de Broglie wavelength of a macroscopic object is ~10⁻³⁵ m, immeasurably smaller than its size, and interaction with the environment destroys coherence (decoherence).
• Q: Which observables can be measured simultaneously and exactly? A: Those whose operators commute, such as L² and L_z, or energy and momentum for a free particle; non-commuting pairs cannot.
• Q: What is the difference between a stationary state and a static system? A: In a stationary state the probability density and expectation values are time-independent, but the particle is not at rest — ⟨p²⟩ and the kinetic energy are non-zero.
• Q: Why does the Compton shift not depend on the incident wavelength? A: Conservation of energy and momentum for a photon–free-electron collision gives Δλ = (h/m₀c)(1 − cos θ), which contains only the angle and the electron's rest mass.
• Q: What does the correspondence principle state? A: Quantum predictions must approach classical ones in the limit of large quantum numbers or ℏ → 0; Ehrenfest's theorem shows expectation values obey Newton's equations.

Common mistakes

• Treating ψ as a measurable field or as the particle's "shape"; only |ψ|² (and matrix elements) are physical.
• Writing E = n²h²/(8mL²) with ℏ in place of h, or mixing E_n = n²π²ℏ²/(2mL²) with the h-form — they are equal, the mix-up is not.
• Forgetting normalisation before computing an expectation value, so ⟨A⟩ comes out wrong by a constant.
• Starting the harmonic oscillator at n = 1; it starts at n = 0, with ½ℏω, while the box starts at n = 1.
• Using non-relativistic λ = h/√(2mE) for electrons of hundreds of keV, or for photons (photons have no rest mass — use λ = hc/E).
• Applying Δx·Δp ≥ ℏ/2 with a factor of h instead of ℏ, or quoting ℏ/2 and then substituting h/4π inconsistently in the same step.
• Saying tunnelling lets the particle "borrow" energy inside the barrier; the energy stays E, the wave function merely decays exponentially there.

Related topics

• Atomic and molecular physics · Nuclear and particle physics · Special relativity · Statistical mechanics and quantum statistics · Solid state physics and band theory · Mathematical physics: differential equations and Fourier analysis.