nLab
Dyson formula

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

$$A=iHdt\in {\Omega }^1(ℝ,𝔲(V))$$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 $dt\in {\Omega }^1(ℝ)$ accordingly the corresponding canonical basis 1-form, the Lie-algebra valued coefficient

$$iH:ℝ\to 𝔲(V)$$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

$$\psi :ℝ\to V.$$\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

$${\partial }_t\psi =iH\psi .$$\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

$$\psi (t)=P\exp ({\int }_{[0,t]}iH(t)dt)\cdot {\psi }_0.$$\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.