Heisenberg's Uncertainty Principle

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Now, a measurement of a particle's wavenumber, , is equivalent to
a measurement of its momentum, , since . Hence,
an uncertainty in of order
translates to
an uncertainty in of order
.
It follows from the above inequality that

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It can be seen from Eqs. (79), (105), and (112)
that at large a particle wavefunction of original width
(at ) spreads out such that its spatial extent becomes

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Figure 8 illustrates a famous thought experiment known as
*Heisenberg's microscope*. Suppose that we try to image an
electron using a simple optical system in which the objective lens is of
diameter and focal-length . (In practice, this would only
be possible using extremely short wavelength light.) It is a
well-known result in optics that such a system has a
minimum angular resolving power of , where
is the wavelength of the light illuminating the electron. If the electron is placed at the focus
of the lens, which is where the minimum resolving power is achieved, then this translates to a uncertainty in the
electron's transverse position of

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(123) |

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Let us now examine Heisenberg's microscope from a quantum mechanical
point of view. According to quantum mechanics, the electron is imaged
when it scatters an incoming photon towards the objective lens.
Let the wavevector of the incoming photon have the
components . See Fig. 8. If the scattered photon
subtends an angle with the center-line of the optical
system, as shown in the figure, then its wavevector is written
. Here,
we are ignoring any wavelength shift of the photon on scattering--*i.e.*,
the magnitude of the -vector is assumed to be the same before and after scattering.
Thus, the change in the -component of the photon's wavevector
is
. This translates to a change
in the photon's -component of momentum of
. By momentum conservation, the
electron's -momentum will change by an equal and opposite
amount. However, can range all the way from to
, and the scattered photon will still be collected by
the imaging system. It follows that the uncertainty in the electron's
momentum is

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(127) |

According to Heisenberg's microscope, the uncertainty principle follows
from two facts. First, it is impossible to measure any property of a microscopic dynamical system without
*disturbing* the system somewhat. Second, particle and light energy and momentum are *quantized*.
Hence, there is a limit to how small we can make the aforementioned
disturbance. Thus, there is an irreducible uncertainty in certain measurements which is a consequence of the act of measurement itself.