# Contents

## Idea

The Dyson formula is an expression for the solution of the Schrödinger equation in time dependent quantum mechanics.

It expresses the parallel transport of the Hamiltonian operator regarded as a Hermitian-operator valued 1-form on the time axis.

## Definition

In time-dependent quantum mechanics dynamics is encoded in a Lie-algebra valued 1-form

$A=iH\phantom{\rule{thinmathspace}{0ex}}dt\in {\Omega }^{1}\left(ℝ,𝔲\left(V\right)\right)$A = i H \,d t \in \Omega^1(\mathbb{R}, \mathfrak{u}(V))

on the real line (time) with values in the Lie algebra of the unitary group on a Hilbert space $V$.

For $t=\mathrm{Id}:ℝ\to ℝ$ the canonical coordinate function and $dt\in {\Omega }^{1}\left(ℝ\right)$ accordingly the corresponding canonical basis 1-form, the Lie-algebra valued coefficient

$iH:ℝ\to 𝔲\left(V\right)$i H : \mathbb{R} \to \mathfrak{u}(V)

of $A$ is called the Hamiltonian operator. If $H$ is a constant function, one speaks of time-independent quantum mechanics.

A state of the system is a function

$\psi :ℝ\to V\phantom{\rule{thinmathspace}{0ex}}.$\psi : \mathbb{R} \to V \,.

A physical state is a solution to the Schrödinger equation

$d\psi +A\cdot \psi =0$d \psi + A \cdot \psi = 0

or equivalently

${\partial }_{t}\psi =iH\psi \phantom{\rule{thinmathspace}{0ex}}.$\partial_t \psi = i H \psi \,.

This differential equation is that which defines the parallel transport of $A$. Its unique solution for given $\psi \left(0\right)$ is written

$\psi \left(t\right)=P\mathrm{exp}\left({\int }_{\left[0,t\right]}iH\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt\right)\cdot {\psi }_{0}\phantom{\rule{thinmathspace}{0ex}}.$\psi(t) = P \exp(\int_{[0,t]} i H(t) \, d t) \cdot \psi_0 \,.

This is called the Dyson formula . In the special case of time-independent quantum mechanics this becomes an ordinary exponential of an ordinary integral.

Revised on September 2, 2010 19:17:53 by Urs Schreiber (131.211.233.99)