Feynman's Principle Quantum amplitude is sum over all possible paths: $$ \langle x_f|e^{-iHt/\hbar}|x_i\rangle = \int \mathcal{D}[x(t)] e^{iS[x]/\hbar} $$ Where $S[x]$ is the classical action: $$ S[x] = \int_0^t L(x, \dot{x}, t') dt' $$ Key Idea Classical mechanics: One path (least action) Quantum mechanics: All paths …

Qubit Superposition of $|0\rangle$ and $|1\rangle$: $$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle $$ Where $|\alpha|^2 + |\beta|^2 = 1$ Quantum Gates Pauli-X (NOT) $$ X = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} $$ Hadamard $$ H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} $$ …

Wave Function State of a quantum system: $$ |\psi\rangle = \sum_i c_i |i\rangle $$ Normalization: $\langle\psi|\psi\rangle = 1$ Born Rule Probability of measuring state $|i\rangle$: $$ P(i) = |\langle i|\psi\rangle|^2 = |c_i|^2 $$ Expectation Value $$ \langle A \rangle = \langle\psi|\hat{A}|\psi\rangle $$ Uncertainty …

Position Operator $$ \hat{x}|\psi\rangle = x|\psi\rangle $$ Momentum Operator $$ \hat{p} = -i\hbar\frac{\partial}{\partial x} $$ Hamiltonian (Energy) $$ \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) $$ Commutator $$ [\hat{x}, \hat{p}] = i\hbar $$ Further Reading Operator (Physics) - Wikipedia

Quantum mechanics basics, common mathematics, and most useful methods.

Time-Dependent $$ i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle $$ Time-Independent For stationary states: $$ \hat{H}|\psi\rangle = E|\psi\rangle $$ 1D Position Space $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi $$ Python (Numerical Solution) 1import numpy as np 2from …