Quantum Mechanics for Mathematicians

Theorem(Heisenberg uncertainty).

〈ψ|Q^2 |ψ〉

〈ψ|ψ〉

〈ψ|P^2 |ψ〉

〈ψ|ψ〉

≥

1

4

Proof.For any realλone has

〈(Q+iλP)ψ|(Q+iλP)ψ〉≥ 0

but, using self-adjointness ofQandP, as well as the relation [Q,P] =ione has

〈(Q+iλP)ψ|(Q+iλP)ψ〉=λ^2 〈ψ|P^2 ψ〉+iλ〈ψ|QPψ〉−iλ〈ψ|PQψ〉+〈ψ|Q^2 ψ〉

=λ^2 〈ψ|P^2 ψ〉+λ(−〈ψ|ψ〉) +〈ψ|Q^2 ψ〉

This will be non-negative for allλif

〈ψ|ψ〉^2 ≤ 4 〈ψ|P^2 ψ〉〈ψ|Q^2 ψ〉

12.5 The propagator in position space

Free particle states with the simplest physical interpretation are momentum
eigenstates. They describe a single quantum particle with a fixed momentum
k′, and this momentum is a conserved quantity that will not change. In the
momentum space representation (see section 11.5) such a time-dependent state
will be given by
ψ ̃(k,t) =e−i^21 mk′^2 tδ(k−k′)

In the position space representation such a state will be given by

ψ(q,t) =

1

√

2 π

e−i

21 mk′^2 t

eik

′q

a wave with (restoring temporarily factors of~and usingp=~k) wavelength
2 π~
p′ and angular frequency

p′^2
2 m~.
As for any quantum system, time evolution of a free particle from time 0 to
timetis given by a unitary operatorU(t) =e−itH. In the momentum space
representation this is just the multiplication operator

U(t) =e−i

21 mk^2 t

In the position space representation it is given by an integral kernel called the
“propagato”r:

Definition(Position space propagator).The position space propagator is the
kernelU(t,qt,q 0 )of the time evolution operator acting on position space wave-
functions. It determines the time evolution of wavefunctions for all timest
by

ψ(qt,t) =

∫+∞

−∞

U(t,qt,q 0 )ψ(q 0 ,0)dq 0 (12.4)

whereψ(q 0 ,0)is the initial value of the wavefunction at time 0.