This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic and potential energy.
Background
Schrödinger's equation Schrödinger's equation, in bra–ket notation, is
i ℏ d d t | ψ ⟩ = H ^ | ψ ⟩ {\displaystyle i\hbar {\frac {d}{dt}}\left|\psi \right\rangle ={\hat {H}}\left|\psi \right\rangle }
where H ^ {\displaystyle {\hat {H}}} is the Hamiltonian operator. The Hamiltonian operator can be written
H ^ = p ^ 2 2 m + V ( q ^ ) {\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\hat {q}})}
where V ( q ^ ) {\displaystyle V({\hat {q}})} is the potential energy, m is the mass and we have assumed for simplicity that there is only one spatial dimension q. The formal solution of the equation is
| ψ ( t ) ⟩ = exp ( − i ℏ H ^ t ) | q 0 ⟩ ≡ exp ( − i ℏ H ^ t ) | 0 ⟩ {\displaystyle \left|\psi (t)\right\rangle =\exp \left(-{\frac {i}{\hbar }}{\hat {H}}t\right)\left|q_{0}\right\rangle \equiv \exp \left(-{\frac {i}{\hbar }}{\hat {H}}t\right)|0\rangle }
where we have assumed the initial state is a free-particle spatial state | q 0 ⟩ {\displaystyle \left|q_{0}\right\rangle } . The transition probability amplitude for a transition from an initial state | 0 ⟩ {\displaystyle \left|0\right\rangle } to a final free-particle spatial state | F ⟩ {\displaystyle |F\rangle } at time T is
⟨ F | ψ ( T ) ⟩ = ⟨ F | exp ( − i ℏ H ^ T ) | 0 ⟩ . {\displaystyle \langle F|\psi (T)\rangle =\left\langle F{\Biggr |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}T\right){\Biggl |}0\right\rangle .}
Path integral formulation The path integral formulation states that the transition amplitude is simply the integral of the quantity
exp ( i ℏ S ) {\displaystyle \exp \left({\frac {i}{\hbar }}S\right)}
over all possible paths from the initial state to the final state. Here S is the classical action. The reformulation of this transition amplitude, originally due to Dirac and conceptualized by Feynman, forms the basis of the path integral formulation.
From Schrödinger's equation to the path integral formulation The following derivation makes use of the Trotter product formula, which states that for self-adjoint operators A and B (satisfying certain technical conditions), we have
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