Mathematical Methods for Physicists: A concise introduction - Site Map

HEISENBERG'S UNCERTAINTY PRINCIPLE
Since g…!† is even, so g…!† sin !x is odd and the second integral on the right hand
side of the last equation is zero, and we have
f …x† ˆ 1 Z r
1
Z
2 1
p g…!† cos !xd! ˆ g…!† cos !xd!:
2
1
Similarly, we can prove Fourier sine transforms by replacing the cosine by the
sine.
0
Heisenberg's uncertainty principle
We have demonstrated in above examples that if f …x† is sharply peaked, then g…!†
is ¯attened, and vice versa. This is a general feature in the theory of Fourier
transforms and has important consequences for all instances of wave propagation.
In electronics we understand now why we use a wide-band ampli®cation in
order to reproduce a sharp pulse without distortion.
In quantum mechanical applications this general feature of the theory of
Fourier transforms is related to the Heisenberg uncertainty principle. We saw
in Example 4.9 that the spread of the Fourier transform in k space (k) times
its spread in coordinate space (a) is equal to 2 …ak 2†. This result is of
special importance because of the connection between values of k and momentum
p: p ˆ pk (where p is the Planck constant h divided by 2). A particle localized in
space must be represented by a superposition of waves with di€erent momenta.
As a result, the position and momentum of a particle cannot be measured simultaneously
with in®nite precision; the product of `uncertainty in the position determination'
and `uncertainty in the momentum determination' is governed by the
relation xp h…apk 2p ˆ h, orxp h; x ˆ a†. This statement is
called Heisenberg's uncertainty principle. If position is known better, knowledge
of the momentum must be unavoidably reduced proportionally, and vice versa. A
complete knowledge of one, say k (and so p), is possible only when there is
complete ignorance of the other. We can see this in physical terms. A wave
with a unique value of k is in®nitely long. A particle represented by an in®nitely
long wave (a free particle) cannot have a de®nite position, since the particle can be
anywhere along its length. Hence the position uncertainty is in®nite in order that
the uncertainty in k is zero.
Equation (4.32) represents Heisenberg's uncertainty principle in a di€erent
form. It states that we cannot know with in®nite precision the exact energy of a
quantum system at every moment in time. In order to measure the energy of a
quantum system with good accuracy, one must carry out such a measurement for
a suciently long time. In other words, if the dynamical state exists only for a
time of order t, then the energy of the state cannot be de®ned to a precision
better than h=t.
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