Associative ionization

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

$i\hslash \frac{d}{dt}|\psi ⟩=\hat{H}|\psi ⟩$

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

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

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

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

$⟨F|\psi (t)⟩=⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}T\right)|0⟩.$

Path integral formulation

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

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

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

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, $x_a$ and $x_b$, separated by a time difference, $\delta t$, 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 from 0 to T into N segments of length

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

The transition amplitude can then be written

$⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}T\right)|0⟩=⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)\cdots \exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|0⟩$.

We can insert the identity

$I=\int dq|q⟩⟨q|$

matrix N-1 times between the exponentials to yield

$⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}T\right)|0⟩=(\prod _{j=1}^{N-1}\int d{q}_j)⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|q_{N-1}⟩⟨q_{N-1}|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|q_{N-2}⟩\cdots ⟨q_1|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|0⟩$.

Each individual transition probability can be written

$⟨q_{j+1}|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|q_j⟩=⟨q_{j+1}|\exp \left(-\frac{i}{\hslash }\frac{{\hat{p}}^2}{2m}\delta t\right)\exp \left(-\frac{i}{\hslash }V\left(q_j\right)\delta t\right)|q_j⟩$.

We can insert the identity

$I=\int \frac{dp}{2\pi }|p⟩⟨p|$

into the amplitude to yield

$⟨q_{j+1}|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|q_j⟩=\exp \left(-\frac{i}{\hslash }V\left(q_j\right)\delta t\right)\int \frac{dp}{2\pi }⟨q_{j+1}|\exp \left(-\frac{i}{\hslash }\frac{p^2}{2m}\delta t\right)|p⟩⟨p|q_j⟩$

$=\exp \left(-\frac{i}{\hslash }V\left(q_j\right)\delta t\right)\int \frac{dp}{2\pi }\exp \left(-\frac{i}{\hslash }\frac{p^2}{2m}\delta t\right)⟨q_{j+1}|p⟩⟨p|q_j⟩$

$=\exp \left(-\frac{i}{\hslash }V\left(q_j\right)\delta t\right)\int \frac{dp}{2\pi }\exp (-\frac{i}{\hslash }\frac{p^2}{2m}\delta t-\frac{i}{\hslash }p(q_{j+1}-q_j))$

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

$⟨p|q_j⟩=\frac{\exp \left(\frac{i}{\hslash }p{q}_j\right)}{\sqrt{\hslash }}$.

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

$⟨q_{j+1}|\exp \left(-\frac{i}{\hslash }\hat{H}\delta t\right)|q_j⟩={\left(\frac{-im}{2\pi \delta t\hslash }\right)}^{\frac{1}{2}}\exp \left[\frac{i}{\hslash }\delta t(\frac{1}{2}m{\left(\frac{q_{j+1}-q_j}{\delta t}\right)}^2-V\left(q_j\right))\right]$

The transition amplitude for the entire time period is

$⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}T\right)|0⟩={\left(\frac{-im}{2\pi \delta t\hslash }\right)}^{\frac{N}{2}}(\prod _{j=1}^{N-1}\int d{q}_j)\exp \left[\frac{i}{\hslash }\sum _{j=0}^{N-1}\delta t(\frac{1}{2}m{\left(\frac{q_{j+1}-q_j}{\delta t}\right)}^2-V\left(q_j\right))\right]$.

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

$⟨F|\exp \left(-\frac{i}{\hslash }\hat{H}T\right)|0⟩=\int Dq(t)\exp \left[\frac{i}{\hslash }S\right]$

where S is the classical action given by

$S=\underset{0}{\overset{T}{\int }}dtL(q(t),\dot{q}(t))$

and L is the classical Lagrangian given by

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

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

$\int Dq(t)=\underset{N\to \mathrm{\infty }}{\lim }{\left(\frac{-im}{2\pi \delta t\hslash }\right)}^{\frac{N}{2}}(\prod _{j=1}^{N-1}\int d{q}_j)$

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

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