Playground
Animated Gaussian wave packet spreading over time in free space. Shows |Psi|^2 and real part of Psi.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Wavefunctionoutput Complex probability amplitude, function of position and time | m^-3/2 | L^-3/2 | 0 – 1 | |
| Reduced Planck constant h / (2*pi) | J*s | M*L^2*T^-1 | 1.054571817e-34 – 1.054571817e-34 | |
| Hamiltonian operator Total energy operator: kinetic + potential | J | M*L^2*T^-2 | 0 – 1e-15 |
Deep dive
Derivation
Postulate a wave description Psi(x,t) with de Broglie p = hbar*k and Einstein E = hbar*omega. A plane wave exp(i(kx - omega*t)) satisfies p -> -i*hbar*d/dx and E -> i*hbar*d/dt when acted upon by these derivative operators. Substituting into the classical E = p^2/(2m) + V gives i*hbar*dPsi/dt = [-hbar^2/(2m)*d^2/dx^2 + V] Psi = H*Psi.
Experimental verification
Every quantum experiment ever done — atomic spectra, electron diffraction, tunneling, STM, quantum dots, Bose-Einstein condensates. Accuracy: QED extends it to parts per trillion for the electron g-2.
Common misconceptions
- Psi is not directly observable — only |Psi|^2 is a probability density
- The 'i' is essential — the equation is first-order in time but complex, not second-order real
- It is NOT a classical wave equation — dispersion is quadratic, not linear
Real-world applications
- Quantum chemistry (DFT, Hartree-Fock)
- Semiconductor band structure calculations
- Quantum computing simulations
- Nuclear reaction modeling
Worked examples
Free particle plane wave
Given:
- V:
- 0
Find: dispersion relation
Solution
omega = hbar*k^2/(2m)
Stationary state separation
Given:
- V:
- V(x)
Find: time-independent form
Solution
H*psi(x) = E*psi(x)
Scenarios
What if…
- scenario:
- What if Psi were real?
- answer:
- The equation would force dPsi/dt = 0 — nothing evolves. Complex amplitudes are essential.
- scenario:
- What if H depended on time?
- answer:
- Non-autonomous evolution — solve with time-ordered exponential U(t) = T*exp(-i*integral(H dt')/hbar).
- scenario:
- What if we measure Psi?
- answer:
- Collapse postulate: Psi projects onto an eigenstate of the measured observable (in Copenhagen interpretation).
Limiting cases
- condition:
- V = 0 (free particle)
- result:
- Plane wave solutions exp(i(kx - omega t))
- explanation:
- With dispersion omega = hbar*k^2/(2m) — quadratic, unlike classical waves.
- condition:
- Stationary state
- result:
- Psi(x,t) = psi(x)*exp(-i*E*t/hbar)
- explanation:
- Time dependence factors out, reducing to the time-independent Schrödinger equation.
- condition:
- Classical limit (hbar → 0)
- result:
- Recovers Hamilton-Jacobi equation
- explanation:
- WKB approximation connects quantum amplitudes to classical trajectories.
Context
Erwin Schrödinger · 1926
Inspired by de Broglie's matter waves, Schrödinger wrote down the wave equation governing non-relativistic quantum mechanics.
Hook
What is the 'F = ma' of quantum mechanics?
The wavefunction evolves under this PDE. How does it generalize Newton's second law for quantum systems?
Dimensions:
- lhs:
- [M*L^2*T^-1]*[T^-1]*[L^-3/2] → [M*L^-1/2*T^-2]
- rhs:
- [M*L^2*T^-2]*[L^-3/2] → [M*L^-1/2*T^-2]
- check:
- Both sides match (energy density^1/2 per unit volume). ✓
Validity: Non-relativistic quantum mechanics (v << c). For relativistic systems use Dirac or Klein-Gordon equations. Fails for high-energy pair production.