Quantum Mechanics for Mathematicians

Recall from one of our basic axioms that time evolution of states is given by the
Schr ̈odinger equation
d
dt

|ψ(t)〉=−iH|ψ(t)〉

(we have set~= 1). We will later more carefully study the relation of this
equation to the symmetry of time translation (the Hamiltonian operatorH
generates an action of the groupRof time translations, just as the operatorQ
generates an action of the groupU(1)). For now though, note that for time-
independent Hamiltonian operatorsH, the solution to this equation is given by
exponentiatingH, with
|ψ(t)〉=U(t)|ψ(0)〉

where

U(t) =e−itH= 1−itH+

(−it)^2

2!

H^2 +···

The commutator of two operatorsO 1 ,O 2 is defined by

[O 1 ,O 2 ] :=O 1 O 2 −O 2 O 1

and such operators are said to commute if [O 1 ,O 2 ] = 0. If the Hamiltonian
operatorH and the charge operatorQcommute, thenQwill also commute
with all powers ofH
[Hk,Q] = 0

and thus with the exponential ofH, so

[U(t),Q] = 0

This condition
U(t)Q=QU(t) (2.1)

implies that if a state has a well-defined valueqjfor the observableQat time
t= 0, it will continue to have the same value at any other timet, since

Q|ψ(t)〉=QU(t)|ψ(0)〉=U(t)Q|ψ(0)〉=U(t)qj|ψ(0)〉=qj|ψ(t)〉

This will be a general phenomenon: if an observable commutes with the Hamil-
tonian observable, one gets a conservation law. This conservation law says that
if one starts in a state with a well-defined numerical value for the observable (an
eigenvector for the observable operator), one will remain in such a state, with
the value not changing, i.e., “conserved”.
When [Q,H] = 0, the groupU(1) is said to act as a “symmetry group” of
the system, withπ(eiθ) the “symmetry transformations”. Equation 2.1 implies
that
U(t)eiQθ=eiQθU(t)

so the action of theU(1) group on the state space of the system commutes with
the time evolution law determined by the choice of Hamiltonian. It is only when
a representation determined byQhas this particular property that the action