The Path Integral Formulation of Quantum Mechanics Shekhar Suresh Chandra * March 15, 2005 Prerequisite: Knowledge of the Lagrangian Formalism (which can be found on my site also). Thirty-one years ago, Dick Feynman told me about his sum over histories version of quantum mechanics. The electron does anything it likes, he said. It just goes in any direction at any speed, . . . however it likes, and then you add up the amplitudes and it gives you the wavefunction. I said to him, Youre crazy. But he wasnt. –Freeman Dyson, 1980 [1] In the classical theory of mechanics, the motion of an object occurs only in is to undergo the least action, i.e. it abides the Hamilton’s Principle of Least Action δS = δ  t2 t1 L(q(t), ˙ q(t))dt =0 (1) where the action is defined by S =  t2 t1 L(q(t), ˙ q(t))dt (2) Quantum Mechanically however, the quantum mechanical object explores every possible path in its motion. Richard Feynman developed this idea to formulate Quantum Mechanics in a different way (by using path integrals) and definition (2) played a critical role. This formulation is useful in the theory of Quantum Fields. Non-relativistic quantum mechanics is governed by the Schrodinger Equation i ∂ ∂t |Ψ(t) > =  H(ˆ p, ˆ q)|Ψ(t) > * Monash University. Email: Shekhar.Chandra@spme.monash.edu.au 1