Dyson formula



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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.


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

A=iHdtΩ 1(,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 VV.

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

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

of AA is called the Hamiltonian operator. If HH is a constant function, one speaks of time-independent quantum mechanics.

A state of the system is a function

ψ:V. \psi : \mathbb{R} \to V \,.

A physical state is a solution to the Schrödinger equation

dψ+Aψ=0 d \psi + A \cdot \psi = 0

or equivalently

tψ=iHψ. \partial_t \psi = i H \psi \,.

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

ψ(t)=Pexp( [0,t]iH(t)dt)ψ 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.

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