Time-dependent Schrödinger equation:

\[i\hbar\,\frac{\partial}{\partial t}\,|\psi(t)\rangle \;=\; \hat H\,|\psi(t)\rangle\]

where:

  • $i$ is the imaginary unit with $i^2=-1$.
  • $h$ is Planck’s constant, defined exactly as $6.62607015\times 10^{-34}\ \mathrm{J\cdot s}$ (SI 2019).
  • $\hbar$ (“h-bar”) is the reduced Planck constant, $\hbar \equiv h/(2\pi) \approx 1.054571817\times 10^{-34}\ \mathrm{J\cdot s}$.
  • $\dfrac{\partial}{\partial t}$ is the time derivative.
  • $\lvert \psi(t) \rangle$ is the (normalized) state vector in a Hilbert space; $\langle \psi(t) \mid \psi(t) \rangle = 1$. In position space, $\psi(\mathbf{x},t)=\langle \mathbf{x}\,\mid\,\psi(t)\rangle$.
  • $\hat H$ is the (self-adjoint) Hamiltonian operator (the generator of time translations) representing total energy.
    • For a single nonrelativistic particle of mass $m$ in potential $V(\mathbf{x},t)$: \(\hat H \;=\; \frac{\hat{\mathbf p}^{\,2}}{2m} + V(\hat{\mathbf x},t), \qquad \hat{\mathbf p} \equiv -\,i\hbar\nabla,\quad \hat{\mathbf x}\equiv \mathbf x.\)
  • Unitary evolution: if $\hat H$ is time-independent, \(|\psi(t)\rangle \;=\; e^{-\,\tfrac{i}{\hbar}\hat H (t-t_0)}\,|\psi(t_0)\rangle .\)

Position representation (single particle):

\[i\hbar\,\frac{\partial}{\partial t}\psi(\mathbf{x},t) \;=\; \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{x},t)\right]\psi(\mathbf{x},t).\]

Probability conservation (continuity equation):

\[\frac{\partial}{\partial t}\,|\psi(\mathbf{x},t)|^2 + \nabla\!\cdot\!\mathbf j(\mathbf{x},t)=0, \qquad \mathbf j(\mathbf{x},t)=\frac{\hbar}{m}\,\mathrm{Im}\!\big(\psi^*\nabla\psi\big).\]

Born rule (projective measurement)

For an observable $\hat A=\sum_a a\,\Pi_a$ with projectors ${\Pi_a}$, \(P(a)\;=\;\langle\psi|\Pi_a|\psi\rangle \qquad\text{(or, for a mixed state $\rho$: $P(a)=\mathrm{Tr}[\rho\,\Pi_a]$).}\)

where:

  • $\Pi_a$ projects onto the eigenspace of outcome $a$; projectors satisfy $\Pi_a\Pi_b=\delta_{ab}\Pi_a$ and $\sum_a \Pi_a=\mathbb I$.
  • $P(a)$ is the probability of obtaining eigenvalue $a$ if the measurement described by ${\Pi_a}$ is performed on the state $\lvert \psi \rangle$ (or $\rho$).
  • Continuous case (e.g., position): the probability density is $p(\mathbf{x},t)=\lvert \psi(\mathbf{x},t) \rvert^2$ with normalization $\int \mathrm{d}^3x\,\lvert \psi(\mathbf{x},t) \rvert^2=1$; the probability to find the particle in $\mathrm{d}^3x$ around $\mathbf{x}$ is $p(\mathbf{x},t)\,\mathrm{d}^3x$.
  • Qubit $Z$ measurement: if $|\psi\rangle=\cos(\tfrac{\theta}{2})|0\rangle+e^{i\phi}\sin(\tfrac{\theta}{2})|1\rangle$, then \(P(0)=\cos^2\!\left(\frac{\theta}{2}\right),\qquad P(1)=\sin^2\!\left(\frac{\theta}{2}\right).\)
  • Copenhagen (collapse): probabilities are fundamental; measurement induces collapse to an eigenstate with Born-rule weights.
  • Bohmian (pilot-wave): dynamics is deterministic (guided trajectories); Born statistics arise from an equilibrium distribution and epistemic ignorance of initial conditions.
  • Everett (Many-Worlds): evolution is unitary; probabilities are branch credences (operational, not ontic dice).
  • Objective collapse (GRW/CSL): modify Schrödinger with stochastic terms; randomness is ontic and, in principle, experimentally testable.

TODO: continue and refine this note!