Path integral formulation: Difference between revisions

Revision as of 02:57, 3 December 2024

Revision as of 03:08, 3 December 2024

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: <math>K(x, y; T) = \langle y; T \mid x; 0 \rangle = \int_{x(0)=x}^{x(T)=y} e^{i S[x]} \,Dx.</math>

This is called the [[propagator]]. To obtain the final state at $y$ we begin with the initial state $\psi_0(x)$ andsimply apply ${{math|''K''(''x'',''y''; ''T'')$}} to the initial state and integrate over {{math|''x''}} resulting in:

: <math>\psi_T(y) = \int_x \psi_0(x) K(x, y; T) \,dx = \int^{x(T)=y} \psi_0(x(0)) e^{i S[x]} \,Dx.</math>