Quantum mechanics

• How to Calculate an Electron’s Expectation Value of Velocity
  For a nonrelativistic electron, use the velocity operator p/m and average it with the wavefunction; a worked Gaussian example gives a numerical result.
• Can Quantum Entanglement Be Used for FTL Communication?
  Entanglement produces correlations across distance, but neither party can control the remote outcomes needed to send a faster-than-light message.
• Can a Local Operation Disentangle Two Parties?
  A local unitary cannot change bipartite entanglement, but a local measurement or noisy channel can destroy it. See the Bell-pair proof.
• How Does Pilot-Wave Theory Explain a Quantum Eraser?
  A quantum eraser is compatible with pilot-wave theory: entanglement hides fringes in the unsorted data, while postselected subsets reveal them.
• How to Distinguish Pure, Mixed, Entangled, and Separable States
  Use density-matrix purity to test pure versus mixed states, then apply entanglement criteria such as partial transpose or witnesses to bipartite states.
• Is the Many-Worlds Interpretation Only a Philosophical Matter?
  Many-Worlds is more than a philosophical label: it gives a physical account of measurement, though standard versions share quantum mechanics’ testable predictions.
• Is the Bohr Radius Deprecated?
  The Bohr radius is not deprecated: it remains a standard physical constant and atomic unit, though the Bohr model is only an approximation.
• Can Three or More Spin-1/2 Particles Form a Singlet?
  A total-spin singlet exists for N spin-1/2 particles exactly when N is even. Learn why, how to construct one, and what changes for electrons.
• Functional Analysis for Quantum Mechanics: What You Need to Know
  Functional analysis supplies quantum mechanics with Hilbert spaces, self-adjoint operators, spectra, and the rigorous treatment of measurement and evolution.
• How Isolated Must a System Be to Avoid Wave-Function Collapse?
  There is no universal isolation threshold for avoiding collapse: interactions cause decoherence, while whether that is actual collapse depends on the quantum.
• How to Derive the Expression for Berry Curvature
  Derive Berry curvature from a normalized eigenstate, then express it as a sum over other bands with energy denominators and matrix elements.
• Can Planck’s Constant Be Derived From Maxwell’s Equations?
  Maxwell’s equations predict classical electromagnetic waves, not Planck’s constant; quantum theory supplies the action scale needed to describe photon energies.
• Can a Digital Computer Simulate Quantum Entanglement to Any Precision?
  A classical digital computer can approximate entangled quantum systems to arbitrary precision in principle, but general simulations can require exponential resources.
• What Is Planck’s Constant, Simply Explained?
  Planck’s constant sets the scale of quantum effects and relates a photon’s energy to its frequency through the equation E = hf.
• Are Coherent States of Light Classical or Quantum?
  Coherent states are quantum states with classical-like field behavior, Poisson photon statistics, and unavoidable quantum fluctuations in measurements.
• What Are the Arguments For and Against Many-Worlds?
  Many-worlds keeps the quantum state evolving without collapse, but its costs include explaining probability, branching, and its vast ontology.
• Tensor Product vs. Cartesian Product in Quantum Systems
  For two quantum subsystems, the joint Hilbert space is a tensor product, not a Cartesian product of state vectors; see dimensions, product states, and entanglement.
• Tensor Product vs. Direct Product for Three Spin-1/2 Particles
  For three spin-1/2 particles, the composite Hilbert space is an eight-dimensional tensor product; total-spin sectors give a direct-sum decomposition.