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 …

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