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