Schrödinger Equation

The Schrödinger equation is the time evolution equation of quantum mechanics. Instead of telling us "which trajectory a particle follows", it gives us a complex wave function ψ: the squared modulus |ψ|² determines the probability distribution, and the phase determines interference and evolution. You can switch between four scenarios below - free wave packet, potential well eigenstate, superposition beats, and barrier tunneling - to see how quantum states spread, superpose, and are shaped by potential energy over time.

Wave Function Time Evolution Born Probability Interpretation Stationary & Superposition States Quantum Tunneling

1. Time-Dependent Schrödinger Equation

iħ ∂ψ/∂t = [ -ħ²/(2m) ∂²/∂x² + V(x) ] ψ
Hamiltonian operator Ĥ = T + V determines the time evolution of quantum states.

The left side is the time rate of change, and the right side is the total energy operator of the system. Given the initial state ψ(x,0) and potential V(x), we can predict subsequent quantum evolution.

2. Stationary State Equation

[ -ħ²/(2m) ∂²/∂x² + V(x) ] φ = Eφ
Eigenstates only accumulate global phase; probability density remains constant.

When the wave function is written as φ(x)e-iEt/ħ, the spatial part satisfies the stationary state equation. It is the core tool for finding energy levels and eigenfunctions.

3. Born Probability Interpretation

P(x) = |ψ(x,t)|²
Measurement results appear randomly according to the probability distribution given by |ψ|².

The wave function itself is not a "physical cloud density", but a probability amplitude. What directly corresponds to observation statistics is its squared modulus.

4. Normalization & Expectation Values

∫ |ψ(x,t)|² dx = 1，⟨x⟩ = ∫ x |ψ|² dx
Total probability is conserved; expectation values describe statistical averages, not single measurement trajectories.

An acceptable quantum state must be normalizable. Expectation values, variances, and uncertainties all come from integrating over the probability density.

Interactive Experiment Area

Main panel shows Re(ψ), Im(ψ) and |ψ|²; bottom panel shows potential energy. You can pause, step forward, or randomly sample a position measurement according to the current distribution.

How to Read a "Wave Function Plot"

The most confusing aspect of quantum visualization is that different levels of information are mixed in the same plot. On this page, the orange line is the real part Re(ψ), blue line is the imaginary part Im(ψ), green fill is the probability density |ψ|², and the bottom potential plot shows how the environment shapes evolution. Real and imaginary parts rotate, interfere, and recombine over time; only the green |ψ|² directly corresponds to measurement statistics.

Start with potential V(x)The potential determines the Hamiltonian, which determines allowed eigenstates, propagation speed, and boundary conditions.
Next examine wave function phaseRe(ψ) and Im(ψ) encode phase information; phase differences determine interference, beating, and reflection/transmission.
Finally check probability density|ψ|² determines the probability distribution for position measurements and is what you can directly compare with experimental statistics.
Don't confuse expectation value with trajectory⟨x⟩ is an average position, not the actual path a particle takes in a single experiment.
This page uses normalized units to clarify structural relationships and core mechanisms, not as a replacement for rigorous numerical solvers.

Why do stationary states "appear stationary"

The global factor of a stationary state is e-iEt/ħ, which only rotates the phase in the complex plane without changing |ψ|². Thus the probability density remains stationary, but the real and imaginary parts continue changing over time.

Why do superposition states "beat"

Multiple energy eigenstates have different phase rotation rates. When they superpose, the relative phases change over time, causing the probability density to redistribute spatially. This is beating and quantum oscillation.

Why tunneling doesn't violate energy conservation

Tunneling is not the particle "borrowing energy temporarily to cross the wall". Instead, the wave function has an exponentially decaying but non-zero extension into the barrier region, thus retaining some transmission probability on the other side of the barrier.

Common Misconceptions

  • Misconception 1: The wave function is a real cloud of matter. More accurately, it is a probability amplitude; observation statistics correspond to |ψ|².
  • Misconception 2: A single measurement always gives the expectation value. Actually, single measurements are random; only the average of many repeated experiments approaches the expectation value.
  • Misconception 3: Potential well eigenstates are completely "stationary". The probability density is stationary, but the complex phase continues evolving.

Recommended Learning Sequence

  • Start with free wave packets: Wave functions propagate and spread; probability density moves and diffuses over time.
  • Then examine infinite potential wells: Understand eigenstates, nodes, and stationary energy levels.
  • Next study superposition states: Understand how relative phases of different energy levels cause probability distributions to breathe and beat.
  • Finally look at tunneling: Connect "classically forbidden regions" with quantum probability extension.