Quantum Mechanics for Mathematicians

The resolution of the identity operator of equation 4.6 here is written

1 =

∫∞

−∞

|q〉〈q|dq=

∫∞

−∞

|k〉〈k|dk

The transformation between the|q〉and|k〉bases is given by the Fourier
transform, which in this notation is

〈k|ψ〉=

∫∞

−∞

〈k|q〉〈q|ψ〉dq

where

〈k|q〉=

1

√

2 π

e−ikq

and the inverse Fourier transform

〈q|ψ〉=

∫∞

−∞

〈q|k〉〈k|ψ〉dk

where

〈q|k〉=

1

√

2 π

eikq

12.4 Heisenberg uncertainty

We have seen that, describing the state of a free particle at a fixed time, one
hasδ-function states corresponding to a well-defined position (in the position
representation) or a well-defined momentum (in the momentum representation).
ButQandPdo not commute, and states with both well-defined position and
well-defined momentum do not exist. An example of a state peaked atq= 0
will be given by the Gaussian wavefunction

ψ(q) =e−α

q^2

2

which becomes narrowly peaked forαlarge. By equation 11.6 the corresponding
state in the momentum space representation is

1

√

α

e−

k 22 α

which becomes uniformly spread out asαgets large. Similarly, asαgoes to zero,
one gets a state narrowly peaked atk= 0 in momentum space, but uniformly
spread out as a position space wavefunction.
For states with expectation value ofQandPequal to zero, the width of
the state in position space can be quantified by the expectation value ofQ^2 ,
and its width in momentum space by the expectation value ofP^2. One has the
following theorem, which makes precise the limit on simultaneously localizability
of a state in position and momentum space