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.

Background

Schrödinger's equation

Schrödinger's equation, in bra–ket notation, is

$i\hbar \frac{d}{dt} |\psi\rangle = \hat H |\psi\rangle$

where $\hat H$ is the Hamiltonian operator. We have assumed for simplicity that there is only one spatial dimension.

The Hamiltonian operator can be written

$\hat H = \frac{\hat{p}^2}{2m} + V(\hat q )$

where $V(\hat q )$ 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

$|\psi(t)\rangle = \exp\left(-\frac{i}{\hbar} \hat H t\right) |q_0\rangle \equiv \exp\left(-\frac{i}{\hbar} \hat H t\right) |0\rangle$

where we have assumed the initial state is a free-particle spatial state $|q_0\rangle$.

The transition probability amplitude for a transition from an initial state $|0 \rangle$ to a final free-particle spatial state $|F \rangle$ at time T is

$\langle F |\psi(t)\rangle = \left \langle F \bigg| \exp\left(- \frac{i}{\hbar} \hat H T\right) \bigg |0 \right \rangle.$

Path integral formulation

The path integral formulation states that the transition amplitude is simply the integral of the quantity

$\exp\left( \frac{i}{\hbar} S \right)$

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[1] and conceptualized by Feynman,[2] forms the basis of the path integral formulation.[3]

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.

$p = m \left(\frac{x_b - x_a}{\delta t}\right)$

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

$\delta t = \frac{T}{N}.$

The transition amplitude can then be written

$\left \langle F \bigg| \exp\left(-\frac{i}{\hbar} \hat H T \right) \bigg|0 \right \rangle = \left \langle F \bigg | \exp\left(-\frac{i}{\hbar} \hat H \delta t \right) \exp\left(-\frac{i}{\hbar} \hat H \delta t \right) \cdots \exp\left(-\frac{i}{\hbar} \hat H \delta t \right) \bigg| 0 \right \rangle.$

We can insert the identity matrix

$I = \int dq |q\rangle \langle q |$

N − 1 times between the exponentials to yield

$\left \langle F \bigg| \exp\left(- \frac{i}{\hbar} \hat H T \right) \bigg| 0 \right \rangle = \left( \prod_{j=1}^{N-1} \int dq_j \right) \left \langle F \bigg | \exp\left(- \frac{i}{\hbar} \hat H \delta t \right) \bigg| q_{N-1} \right \rangle \left \langle q_{N-1} \bigg | \exp\left(- \frac{i}{\hbar} \hat H \delta t \right) \bigg | q_{N-2} \right \rangle \cdots \left \langle q_{1} \bigg | \exp\left(- \frac{i}{\hbar} \hat H \delta t \right) \bigg | 0 \right \rangle.$

Each individual transition probability can be written

$\left \langle q_{j+1} \bigg| \exp\left(- \frac{i}{\hbar} \hat H \delta t \right) \bigg|q_j \right \rangle = \left \langle q_{j+1} \Bigg| \exp\left( {- {i \over \hbar } { {\hat p}^2 \over 2m} \delta t} \right) \exp\left( {- {i \over \hbar } V \left( q_j \right) \delta t} \right)\Bigg | q_j \right \rangle.$

We can insert the identity

$I = \int { dp \over 2\pi } |p\rangle \langle p |$

into the amplitude to yield

\begin{align} \left \langle q_{j+1} \bigg| \exp\left(- \frac{i}{\hbar} \hat H \delta t \right) \bigg|q_j \right \rangle &= \exp\left(- \frac{i}{\hbar} V \left( q_j \right) \delta t \right) \int \frac{dp}{2\pi} \left \langle q_{j+1} \bigg| \exp\left(-\frac{i}{\hbar} \frac{p^2}{2m} \delta t \right) \bigg | p \right \rangle \langle p |q_j\rangle \\ &= \exp\left(-\frac{i}{\hbar} V \left( q_j \right) \delta t \right) \int \frac{dp}{2\pi} \exp\left(-\frac{i}{\hbar} \frac{p^2}{2m} \delta t \right) \left \langle q_{j+1} |p \right \rangle \left \langle p |q_j \right \rangle \\ &= \exp \left(- \frac{i}{\hbar} V \left( q_j \right) \delta t \right) \int \frac{dp}{2\pi} \exp\left(-\frac{i}{\hbar} \frac{p^2}{2m} \delta t -\frac{i}{\hbar} p \left( q_{j+1} - q_{j} \right) \right) \end{align}

where we have used the fact that the free particle wave function is

$\langle p |q_j\rangle = \frac{\exp\left(\frac{i}{\hbar} p q_{j} \right)}{\sqrt{\hbar}}$.

The integral over p can be performed (see Common integrals in quantum field theory) to obtain

$\left \langle q_{j+1} \bigg| \exp\left(-\frac{i}{\hbar} \hat H \delta t \right) \bigg|q_j \right \rangle = \left( {-i m \over 2\pi \delta t \hbar } \right)^{1\over 2} \exp\left[ {i\over \hbar} \delta t \left( {1\over 2} m \left( {q_{j+1}-q_j \over \delta t } \right)^2 - V \left( q_j \right) \right) \right]$

The transition amplitude for the entire time period is

$\left \langle F \bigg| \exp\left(- \frac{i}{\hbar} \hat H T \right) \bigg|0 \right \rangle = \left( {-i m \over 2\pi \delta t \hbar } \right)^{N\over 2} \left( \prod_{j=1}^{N-1} \int dq_j \right) \exp\left[ {i\over \hbar} \sum_{j=0}^{N-1} \delta t \left( {1\over 2} m \left( {q_{j+1}-q_j \over \delta t } \right)^2 - V \left( q_j \right) \right) \right].$

If we take the limit of large N the transition amplitude reduces to

$\left \langle F \bigg| \exp\left( {- {i \over \hbar } \hat H T} \right) \bigg |0 \right \rangle = \int Dq(t) \exp\left[ {i\over \hbar} S \right]$

where S is the classical action given by

$S = \int_0^T dt L\left( q(t), \dot{q}(t) \right)$

and L is the classical Lagrangian given by

$L\left( q, \dot{q} \right) = {1\over 2} m {\dot{q}}^2 - V (q)$

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

$\int Dq(t) = \lim_{N\to\infty}\left(\frac{-i m}{2\pi \delta t \hbar} \right)^{\frac{N}{2}} \left( \prod_{j=1}^{N-1} \int dq_j \right)$

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.

References

1. ^ Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, Fourth Edition. Oxford. ISBN 0-19-851208-2.
2. ^ Richard P. Feynman (1958). Feynman's Thesis: A New Approach to Quantum Theory. World Scientific. ISBN 981-256-366-0.
3. ^ A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6.