The Schrödinger Equation | mysimulator.uk

Q: What does the wave function Ψ actually represent physically?

Ψ itself is not directly observable — only |Ψ|², the Born rule, has physical meaning as the probability density of finding the particle at a given position. This was a deliberate, radical departure from classical determinism: quantum mechanics predicts probabilities, not exact trajectories.

Q: Why can a particle have a nonzero minimum energy in a box?

Because the boundary conditions ψ(0)=ψ(L)=0 only admit standing-wave solutions ψ_n = √(2/L)·sin(nπx/L) for integer n ≥ 1, and n=0 is not a valid solution (it is identically zero everywhere). The lowest allowed state n=1 has energy E₁ = π²ℏ²/(2mL²) > 0 — this zero-point energy is a direct consequence of the Heisenberg uncertainty principle, which forbids a particle from being perfectly at rest in a confined region.

The wave function and the Born rule

Quantum mechanics describes a particle not with a position and velocity but with a complex-valued wave function Ψ(x,t). Ψ itself is not observable — only |Ψ|², the probability density of finding the particle in a small interval dx, has direct physical meaning:

ρ(x,t) = |Ψ(x,t)|²
∫ |Ψ|² dx = 1          // normalisation — the particle is somewhere
// Born rule (Max Born, 1926): |Ψ|² is a probability density, not a physical wave

If Ψ₁ and Ψ₂ are both valid states, so is any linear combination c₁Ψ₁+c₂Ψ₂ — superposition, the source of every quantum interference effect.

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The equation itself

Schrödinger's 1926 equation governs how Ψ evolves in time under a Hamiltonian Ĥ combining kinetic and potential energy:

iℏ ∂Ψ/∂t = ĤΨ = [ −ℏ²/(2m)·∂²/∂x² + V(x) ] Ψ
// when V is time-independent, solutions separate:
Ψ(x,t) = ψ(x)·e^(−iEt/ℏ)
// leaving the time-independent Schrödinger equation (TISE), an eigenvalue problem:
−ℏ²/(2m)·d²ψ/dx² + V(x)ψ = Eψ

Solving the TISE means finding the Hamiltonian's eigenvalues E (the allowed energies) and eigenfunctions ψ(x) (the stationary states) — everything about a time-independent quantum system reduces to this one eigenvalue problem.

Particle in a box: the simplest exact solution

Infinite walls at x=0 and x=L force ψ(0)=ψ(L)=0, giving standing-wave solutions with quantised energy:

ψ_n(x) = √(2/L)·sin(nπx/L),   n = 1, 2, 3, …
E_n = n²π²ℏ²/(2mL²) = n²·E₁
E₁ = π²ℏ²/(2mL²)          // zero-point energy — E=0 is forbidden

Only discrete energies are allowed; the lowest one E₁ is strictly positive, a direct signature of the Heisenberg uncertainty principle forbidding a confined particle from sitting perfectly still. This exact model approximates conjugated π-electron systems in chemistry and quantum-well semiconductor lasers.

Quantum tunnelling

A particle with energy E < V₀ can still penetrate a finite barrier — a purely quantum effect with zero classical analogue, because the Schrödinger equation still has a valid (exponentially decaying) solution inside the classically forbidden region:

T = [ 1 + V₀²·sinh²(κa) / (4E(V₀−E)) ]⁻¹     // exact, rectangular barrier of width a
κ = √(2m(V₀−E)) / ℏ                          // imaginary wavenumber inside barrier
// WKB approximation for slowly-varying V(x):
T ≈ exp( −2 ∫ κ(x) dx ),  κ(x) = √(2m[V(x)−E]) / ℏ

Tunnelling underlies scanning tunnelling microscopy, flash memory cells, tunnel diodes, alpha decay, and proton-proton fusion in the Sun's core, which proceeds at temperatures roughly ten times lower than classical physics alone would require.

Wave packets and uncertainty

A localised particle is modelled as a Gaussian wave packet — a superposition of plane waves — that spreads over time because each momentum component travels at its own speed v = p/m:

Δx · Δp ≥ ℏ/2                       // Heisenberg uncertainty principle
σ(t) = σ√(1 + (ℏt/(2mσ²))²)          // packet width grows with time

This is not a statement about measurement disturbance — it is an intrinsic property of every wave-like system: a narrow position distribution necessarily carries a broad spread of momentum. This site's wave packet simulation advances the state with a split-step Fourier method, alternating updates between position and momentum space each time step.

Frequently asked questions

What does the wave function Ψ actually represent physically?

Ψ itself is not directly observable — only |Ψ|², the Born rule, has physical meaning as the probability density of finding the particle at a given position. This was a deliberate, radical departure from classical determinism: quantum mechanics predicts probabilities, not exact trajectories.

Why can a particle have a nonzero minimum energy in a box?

Because the boundary conditions ψ(0)=ψ(L)=0 only admit standing-wave solutions ψ_n = √(2/L)·sin(nπx/L) for integer n ≥ 1, and n=0 is not a valid solution (it is identically zero everywhere). The lowest allowed state n=1 has energy E₁ = π²ℏ²/(2mL²) > 0 — this zero-point energy is a direct consequence of the Heisenberg uncertainty principle, which forbids a particle from being perfectly at rest in a confined region.

How can a particle tunnel through a barrier it doesn't have enough energy to climb?

Inside a classically forbidden region (V > E) the Schrödinger equation still has a valid solution — an exponentially decaying wave function rather than an oscillating one. If the barrier is thin enough, that decaying amplitude is still nonzero on the far side, giving a real, computable probability of transmission with no classical analogue at all. It underlies scanning tunnelling microscopy, tunnel diodes and stellar nuclear fusion.

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