Playground
Time-independent Schrödinger equation: standing wave solutions psi_n in an infinite well with energy levels.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Energy eigenvalueoutput Allowed total energy of the quantum state | J | M·L²·T⁻² | -1e-17 – 1e-17 | |
| Particle mass Mass of the quantum particle | kg | M | 1e-31 – 1e-25 | |
| Potential energy External potential energy field | J | M·L²·T⁻² | -1e-17 – 1e-17 |
Deep dive
Derivation
Starting from de Broglie's matter waves and demanding linearity + plane-wave solutions e^{i(kx − ωt)} that respect E = ℏω and p = ℏk, one constructs an operator equation iℏ ∂ψ/∂t = Ĥψ with Ĥ = −ℏ²∇²/(2m) + V. Separating variables ψ(x,t) = ψ(x)e^{−iEt/ℏ} yields the time-independent form Ĥψ = Eψ.
Experimental verification
The hydrogen spectrum, predicted exactly by solving the Schrödinger equation in a Coulomb potential, matches observation to ~5 decimal places (with QED corrections beyond). Quantum tunneling rates in α-decay (Gamow), STM, and Josephson junctions all confirm the equation's predictions.
Common misconceptions
- ψ itself is not directly observable — only |ψ|² gives probability density.
- The equation is deterministic; randomness enters only at measurement (Born rule).
- It is non-relativistic — invalid for fast electrons in heavy atoms; use Dirac equation there.
Real-world applications
- All of computational chemistry (DFT, Hartree–Fock, coupled cluster).
- Semiconductor band structure and transistor design.
- Quantum dots, lasers, LEDs, and photovoltaic devices.
- MRI and NMR rely on quantum spin states governed by Ĥ.
Worked examples
Electron in a 1 nm infinite well
Given:
- L:
- 1e-9
- m_e:
- 9.109e-31
- n:
- 1
Find: ground-state energy E_1
Solution
E_1 = π²ℏ²/(2m_e L²) ≈ 6.0×10⁻²⁰ J ≈ 0.376 eV
Harmonic oscillator ground-state energy
Given:
- ω:
- 100000000000000
Find: E_0
Solution
E_0 = ℏω/2 ≈ 5.27×10⁻²¹ J
Scenarios
What if…
- scenario:
- What if the well were 10× wider?
- answer:
- E_1 ∝ 1/L² → energy drops by 100×. Larger boxes (quantum dots) emit redder light — basis of size-tuned LEDs.
- scenario:
- What if V were a delta function attractive potential?
- answer:
- Exactly one bound state exists in 1D, with E = −mα²/(2ℏ²). Used as a toy model for impurity states in solids.
- scenario:
- What if we used the Schrödinger equation for a relativistic electron in hydrogen?
- answer:
- Predictions miss fine structure (~10⁻⁴ corrections), Lamb shift, and antiparticles. Dirac's equation fixes these.
Limiting cases
- condition:
- V = 0 (free particle)
- result:
- ψ ∝ e^{ikx}, E = ℏ²k²/(2m)
- explanation:
- Plane-wave solutions form a continuum of energies — no quantization without confinement.
- condition:
- V → ∞ outside region
- result:
- Discrete bound states
- explanation:
- Confinement quantizes the energy spectrum (e.g., particle in a box: E_n = n²π²ℏ²/(2mL²)).
- condition:
- ℏ → 0
- result:
- Classical Hamilton–Jacobi equation
- explanation:
- Quantum mechanics smoothly reduces to classical mechanics in the small-ℏ limit (WKB approximation).
Context
Erwin Schrödinger · 1926
Schrödinger wrote down his wave equation during a holiday in the Alps, founding wave mechanics.
Hook
What equation governs the ghostly wavefunction of every electron in the universe?
Find the ground-state energy of an electron trapped in a 1 nm infinite square well.
Dimensions: [ℏ²/(2m)·∇²ψ] = (M·L²·T⁻¹)²/(M·L²) · ψ = M·L²·T⁻²·ψ = [E·ψ] ✓
Validity: Non-relativistic single-particle quantum mechanics. For v approaching c, replace with the Dirac or Klein–Gordon equation. Many-body systems require the full N-particle Schrödinger equation in 3N-dimensional configuration space.