Schrödinger equation
The Schrödinger equation is the differential equation for quantum time evolution. In Susskind’s structure, it is not the starting point; it is the continuous-time form of unitary evolution generated by the Hamiltonian.
Abstract form
iħ d|Ψ⟩/dt = H|Ψ⟩This equation says that the Hamiltonian H determines how the state-vector changes with time.
Position-space particle form
For a one-dimensional particle with Hamiltonian H = P²/(2m) + V(X), representing the state by ψ(x,t) gives:
iħ ∂ψ/∂t = [-(ħ²/2m) ∂²/∂x² + V(x)] ψThe kinetic term differentiates the wavefunction; the potential term multiplies it.
What it predicts
Given an initial wavefunction and a Hamiltonian, the equation predicts the later wavefunction. Measurement probabilities are then computed with the Born rule.
Common pitfalls
- Do not memorize the particle equation as if it were the whole subject. It is one representation of a more general state-vector equation.
- Do not forget that the Hamiltonian comes from the physical system.
- Do not confuse the equation’s deterministic evolution of ψ with deterministic individual measurement outcomes.