# 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

$A = i H \,d t \in \Omega^1(\mathbb{R}, \mathfrak{u}(V))$

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

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

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

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

$\psi : \mathbb{R} \to V \,.$

A physical state is a solution to the Schrödinger equation

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

or equivalently

$\partial_t \psi = i H \psi \,.$

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

$\psi(t) = P \exp(\int_{[0,t]} i H(t) \, d t) \cdot \psi_0 \,.$

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)