Interactive visualization of quantum superposition, time evolution, and wave function measurement collapse in various potential wells
A quantum system can exist in a superposition of multiple eigenstates simultaneously: Ψ(x,0) = c₁ψ₁ + c₂ψ₂ + ... The coefficients cₙ are complex numbers with |cₙ|² representing the probability of finding the system in state n upon measurement. This is fundamentally different from classical physics — the electron is not in state 1 OR state 2, but in BOTH simultaneously. The phase relationships between cₙ create interference patterns in the probability density |Ψ|².
Each eigenstate evolves in time with a phase factor: ψₙ(x,t) = ψₙ(x)e^{-iEₙt/ℏ}. The total wavefunction Ψ(x,t) = Σcₙψₙ(x)e^{-iEₙt/ℏ} is a superposition of phase-rotated eigenstates. The probability density |Ψ|² oscillates because the relative phases between different energy components change at frequencies ωₙₘ = (Eₙ-Eₘ)/ℏ. If only one eigenstate is present, |Ψ|² is static — all dynamics come from interference between different energy levels.
Max Born's 1926 rule: the probability of finding a particle at position x is |Ψ(x,t)|² dx. When a measurement is performed, the wave function 'collapses' — the system instantaneously jumps to one of the eigenstates, with probability |cₙ|². This is not a smooth process but a discontinuous jump. Whether collapse is physical (Copenhagen) or apparent (Many-Worlds, decoherence) remains debated, but the statistical predictions are identical and extraordinarily well-tested.
The simplest quantum system: a particle confined between two impenetrable walls at x=0 and x=L. Wave functions ψₙ(x) = √(2/L)sin(nπx/L) vanish at the boundaries. Energies Eₙ = n²π²ℏ²/(2mL²) increase quadratically — the spacing grows with n. The ground state (n=1) has nonzero energy (zero-point energy) — the particle cannot be at rest. This is a direct consequence of the uncertainty principle: confining the particle to a small region requires large momentum uncertainty, hence kinetic energy.
Unlike the infinite well, the walls have finite height V₀. Wave functions decay exponentially in the classically forbidden region |x| > L/2: ψ ~ e^{-κx} where κ = √(2m(V₀-E)/ℏ²). This tunneling into the barrier is purely quantum. Key features: (1) Only finitely many bound states exist; (2) Energies are lower than infinite-well counterparts because the wavefunction spreads beyond the well; (3) Higher levels approach the infinite-well limit.
V(x) = ½mω²x², the most important potential in physics. Any smooth potential near its minimum is approximately harmonic. Eigenstates use Hermite polynomials: ψₙ(x) = NₙHₙ(ξ)e^{-ξ²/2}. Unique property: equally spaced energy levels Eₙ = (n+½)ℏω. This means superpositions oscillate coherently — the probability density is exactly periodic with period T = 2π/ω. Applications: molecular vibrations, phonons, quantum fields, laser physics.
Two finite wells separated by a barrier. The key phenomenon is quantum tunneling through the barrier: a particle localized in one well can spontaneously appear in the other. The lowest two energy states are split by a small energy ΔE — one symmetric and one antisymmetric. A particle starting in one well oscillates between wells with frequency ΔE/ℏ. This models: ammonia molecule inversion, superconducting qubits, proton tunneling in hydrogen bonds, and chemical isomerization.
Quantum measurement is fundamentally different from classical observation. Before measurement, the system exists in superposition Ψ = Σcₙψₙ. Upon measurement of energy, the system collapses to a single eigenstate ψₖ with probability |cₖ|². After collapse, all interference between energy levels vanishes — the system is now in a definite energy state. Repeating the same measurement yields the same result with certainty. This irreversible projection is the most controversial aspect of quantum mechanics.
Copenhagen: collapse is a real physical process. Many-Worlds: no collapse — the universe splits into branches. Decoherence: interaction with the environment rapidly suppresses interference, making collapse effectively irreversible. QBism: quantum states represent an observer's knowledge. All interpretations make identical experimental predictions — the differences are philosophical. The practical consensus: shut up and calculate.
Schrodinger's 1935 thought experiment: a cat in a sealed box with a radioactive atom, Geiger counter, and vial of poison. If the atom decays, the poison is released. Before opening the box, is the cat alive, dead, or in superposition |alive⟩ + |dead⟩? Modern resolution: decoherence from the environment collapses the cat's state almost instantaneously — macroscopic superposition is impossible in practice.