Quantum Mechanics

A New Kind of Physics

Quantum mechanics is the most accurate physical theory ever devised. It describes the behavior of electrons, photons, atoms, and all other particles at the quantum scale. It is also the most conceptually radical departure from classical physics.

Classical physics assumed that particles have definite positions and velocities at all times, and that measurement merely reads off these pre-existing values. Quantum mechanics says otherwise: particles do not have definite properties until they are measured. The act of measurement fundamentally changes what is observed.

Wave-Particle Duality

Light sometimes behaves like a wave (interference, diffraction) and sometimes like a particle (the photoelectric effect). Electrons sometimes behave like particles (they have mass and charge) and sometimes like waves (electron diffraction patterns). This is wave-particle duality.

The de Broglie wavelength associated with a particle of momentum p is λ = h/p. Every particle has a corresponding wave. This is strange but experimentally confirmed.

The Wavefunction

In quantum mechanics, a particle is described by a wavefunction ψ(x, t). The wavefunction contains all the information about the particle. Its absolute square |ψ|² gives the probability density of finding the particle at position x at time t.

The wavefunction is not a physical wave in space — it is a probability amplitude. This is a new kind of mathematical object with no classical analog.

The Schrödinger Equation

The time evolution of the wavefunction is governed by the Schrödinger equation. It is the quantum analog of Newton's second law. For a particle in a potential V(x), the time-dependent Schrödinger equation is:

iℏ ∂ψ/∂t = (-ℏ²/2m) ∂²ψ/∂x² + V(x)ψ

You will solve this for simple potentials: the infinite square well, the harmonic oscillator, and the hydrogen atom.

Superposition and Measurement

A quantum system can be in a superposition of multiple states simultaneously. When you measure it, the superposition collapses to one outcome, with probabilities given by the wavefunction. This is the measurement problem — one of the deepest unresolved questions in physics.

The Uncertainty Principle

Heisenberg's uncertainty principle states that the more precisely you know a particle's position, the less precisely you can know its momentum, and vice versa: Δx Δp ≥ ℏ/2. This is not a limitation of our measuring instruments — it is a fundamental feature of nature.

Tunneling, Spin, and Entanglement

Quantum particles can tunnel through energy barriers that classical particles cannot penetrate — this is why nuclear fusion occurs in stars. Electrons and other particles carry intrinsic angular momentum called spin. Entangled particles share a quantum state even when separated by large distances — a phenomenon Einstein called "spooky action at a distance."