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Quantum Amplitude

The quantum amplitude, ∣Ψ(x,t)∣|\Psi(x, t)|^2, is the probability density of finding a particle at position \(x) at time \(t), arising from the path integral.
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The statement of the theorem

The probability density of finding a particle at position xbx_b at time tbt_b, given it started at xax_a at tat_a, is given by the square modulus of the propagator, which is calculated via the path integral: \nP(xb,tb∣xa,ta)=∣⟨xb,tb∣xa,ta⟩∣2=∣∫Dx(t)eiℏS[x(t)]∣2P(x_b, t_b | x_a, t_a) = |\langle x_b, t_b | x_a, t_a \rangle|^2 = \left| \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]} \right|^2