ft1ln05_08

# ft1ln05_08 - MIT OpenCourseWare http/ocw.mit.edu 8.323...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth March 23, 2008 QUANTUM MECHANICS AND PATH INTEGRALS The goal of this section is to derive the path integral formulation of quantum mechanics. Consider first a free particle, moving in one dimension: 2 H = p . (5.1) 2 m The evolution of a state is described by applying the operator U ( t f ) ≡ e − iHt f /h ¯ . Let U fi ≡ x f e − iHt f /h ¯ x i . (5.2) To develop a path integral expression for this matrix element, begin by dividing the interval ≤ t ≤ t f into N + 1 equal steps, so ( N + 1)∆ t = t f : 1 2 3 N | | | | | | ←→ | | | | ∆ t t f Now express the evolution operator e − iHt f /h ¯ as the product of an evolution operator for each interval ∆ t : N +1 e − iHt f /h ¯ = e − iH ∆ t/h ¯ . (5.3) Then insert a complete set of intermediate states between each factor, using ∞ 1 = dx | x x | . (5.4) −∞ Calling the variables of integration x 1 , x 2 , . . ., x N , ∞...
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ft1ln05_08 - MIT OpenCourseWare http/ocw.mit.edu 8.323...

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