Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
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.
Schrödinger's equation, in bra–ket notation, is
where is the Hamiltonian operator. We have assumed for simplicity that there is only one spatial dimension.
The Hamiltonian operator can be written
where 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
where we have assumed the initial state is a free-particle spatial state .
The transition probability amplitude for a transition from an initial state to a final free-particle spatial state at time T is
Path integral formulation
The path integral formulation states that the transition amplitude is simply the integral of the quantity
over all possible paths from the initial state to the final state. Here S is the classical action.
From Schrödinger's equation to the path integral formulation
Note: the following derivation is heuristic (it is valid in cases in which the potential, V(q), commutes with the momentum, p). Following Feynman, this derivation can be made rigorous by writing the momentum, p, as the product of mass, m, and a difference in position at two points, xa and xb, separated by a time difference, δt, thus quantizing distance.
Note 2: There are two errata on page 11 in Zee, both of which are corrected here.
We can divide the time interval [0, T] into N segments of length
The transition amplitude can then be written
We can insert the identity matrix
N − 1 times between the exponentials to yield
Each individual transition probability can be written
We can insert the identity
into the amplitude to yield
where we have used the fact that the free particle wave function is
The integral over p can be performed (see Common integrals in quantum field theory) to obtain
The transition amplitude for the entire time period is
If we take the limit of large N the transition amplitude reduces to
where S is the classical action given by
and L is the classical Lagrangian given by
Any possible path of the particle, going from the initial state to the final state, is approximated as a broken line and included in the measure of the integral
This expression actually defines the manner in which the path integrals are to be taken. The coefficient in front is needed to ensure that the expression has the correct dimensions, but it has no actual relevance in any physical application.
This recovers the path integral formulation from Schrödinger's equation.
- Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, Fourth Edition. Oxford. ISBN 0-19-851208-2.
- Richard P. Feynman (1958). Feynman's Thesis: A New Approach to Quantum Theory. World Scientific. ISBN 981-256-366-0.
- A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6.