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 …
Read MoreWave 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 …
Read MorePosition 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
Read MoreTime-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 …
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