Quantum physics

1. Superposition • A quantum system can exist in a linear combination of states (e.g. Schrödinger’s cat). • Mathematically: |ψ⟩ = c₁|ψ₁⟩ + c₂|ψ₂⟩. (Schrödinger 1935) 2. Wave–Particle Duality • Particles (electrons, photons) display both localized (“particle”) and delocalized (“wave”) behavior. • Illustrated by double‐slit experiments. (de Broglie 1924; Davisson & Germer 1927) 3. Uncertainty Principle • Certain pairs of observables (e.g. position x and momentum p) cannot be simultaneously known with arbitrary precision: Δx·Δp ≥ ħ/2. (Heisenberg 1927) 4. Measurement Problem • Upon measurement, a superposed state appears to “collapse” to a definite outcome. • Raises questions about the role of the observer and the ontology of the wavefunction. (von Neumann 1932; Wigner 1961) 5. Entanglement and Non-locality • Two or more particles can share correlations that defy classical explanation. • Einstein–Podolsky–Rosen (EPR) argued this suggests “elements of reality” missing from quantum theory. • Bell’s theorem (1964) and subsequent experiments (Aspect 1982) show no local hidden‐variable theory can reproduce all quantum predictions. Philosophical Interpretations • Copenhagen (Bohr): Complementarity and observer-dependent phenomena. • Many-Worlds (Everett 1957): Universal wavefunction never collapses; all outcomes occur in branching worlds. • Bohmian Mechanics (Bohm 1952): Particles have definite positions guided by a “pilot wave.” • Quantum Bayesianism (QBism): Wavefunction reflects an agent’s personal degrees of belief. Key References – Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik” (1927) – Schrödinger, E. “Die gegenwärtige Situation in der Quantenmechanik” (1935) – Bell, J. S. “On the Einstein Podolsky Rosen Paradox” (1964)

1. Superposition • Stern–Gerlach with spin-½ particles: before measurement, the beam is in |↑⟩+|↓⟩ and only “chooses” one axis on detection. • Schrödinger’s cat: a radioactive decay both “decayed” and “undecayed” until an observer opens the box. (Schrödinger 1935) 2. Wave–Particle Duality • Electron double‐slit: individual electrons form an interference pattern over time, yet each hits the screen at a single point. (Davisson & Germer 1927) • Single‐photon Mach–Zehnder interferometer: behaves like a wave (interference) unless a detector is placed in one arm. 3. Uncertainty Principle • Electron diffraction by a narrow slit: narrower slit (Δx↓) ⇒ wider spread on the screen (Δp↑). • Homodyne detection in quantum optics: precise phase (quadrature) measurement increases photon‐number uncertainty. 4. Measurement Problem • Wigner’s friend: an observer inside a sealed lab sees a definite outcome, while an outside “Wigner” assigns a superposed state to the entire lab. (Wigner 1961) • Quantum Zeno effect: rapid repeated measurements “freeze” a system’s evolution, highlighting the role of measurement. 5. Entanglement and Non‐locality • Bell test experiments (Aspect 1982): polarization‐entangled photons violate Bell inequalities, ruling out local hidden variables. • Quantum teleportation: uses an entangled pair and classical communication to transfer an unknown quantum state. (Bennett et al. 1993) Philosophical Interpretation Examples • Copenhagen: Complementarity in Bohr’s hydrogen‐spectral‐line experiments—wave and particle descriptions are mutually exclusive but jointly exhaustive. • Many-Worlds: Quantum computing’s parallelism is often portrayed as computation across branching worlds. (Everett 1957) • Bohmian Mechanics: “Trajectories” reconstructed in weak‐measurement double‐slit setups mimic pilot‐wave predictions. (Kocsis et al. 2011) • QBism: Photon‐counting statistics in quantum tomography are viewed as an agent’s personal probability assignments rather than objective wavefunction collapse.

Definition • Intrinsic angular momentum: a quantum degree of freedom with magnitude √(s(s+1))ħ (here s=½), so each measurement along any axis yields ±ħ/2. • Fundamentally two-level (“qubit-like”) system, not arising from literal spinning. Experimental Signature • Stern–Gerlach (1922): silver atoms deflected into two discrete spots, revealing ±½ħ projections. (Stern & Gerlach 1922) • Electron spin resonance (ESR): transitions between spin-up and spin-down in a magnetic field produce characteristic absorption frequencies. Physical Consequences • Magnetic moment μ = g(eħ/2m): underpins electron paramagnetism, NMR/MRI imaging, and spintronics devices. • Pauli exclusion principle: no two electrons (spin-½ fermions) occupy the same quantum state ⇒ atomic shell structure and chemistry. (Pauli 1925) • Fine and hyperfine spectral splitting in atoms: electron spin coupling to orbital motion and nuclear spin. Technological Applications • Quantum computing: electron or nuclear spins serve as qubits with long coherence times. • Spintronics: manipulation of spin currents for nonvolatile memory (MRAM) and logic. References • Stern, W. & Gerlach, O. “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld” (1922) • Pauli, W. “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren” (1925)