Wave Mechanics 197
Let us now consider the limitations the uncertainty principle
imposes for the case of an electron in an atom. Planck’s constant will no
longer be such a small number. Instead of distances, like 3 meters for the
automobile, we must now consider distances of the order of 10-8 cm.
Instead of a mass of 1000 kg the mass of the electron is 0.9 × 10-30 kg,
and hence, its momentum is considerably less than that of the auto-
mobile. For an electron with a velocity of 0.1c the product of its position
times its momentum (10-8 cm × 0.9 × 10-30 kg × 0.1 × 3 10^8 m/sec) is
approximately 27 × 10-34 kgm^2 /sec or just 4 h. It is clear that the
uncertainty principle imposes severe limitations on how accurately and
the momentum and position of an electron may be determined since the
uncertainty is the same order of magnitude as the quantities to be
measured.
Let us consider physically what is involved in determining the
position and momentum of an electron. In actuality, it is not much
different than measuring the position and velocity of an automobile by
crashing another automobile into it. There are no particles smaller than
an electron. Therefore, if we wish to detect the electron using another
particle the best we can do is to use another electron. We cannot chop an
electron into a thousand pieces and use a tiny chunk of an electron as a
detector. We are obliged to use another electron or a larger particle.
The only other alternative is to use a photon. This presents a problem
as well because the photon also carries momentum. This problem can be
minimized by choosing to use a low energy and consequently a low
momentum photon. The only difficulty with a low momentum photon is
the fact that it will have a large wavelength, λ, since λ = h/p.
The size of a photon with wavelength λ is at least equal to λ and
hence, the detection of the electron’s position with a photon of
wavelength λ will automatically introduce an uncertainty of at least
∆x = λ. The uncertainty in momentum inherent in the measurement is
just the momentum of the photon, hence ∆p = h/λ. The product of the
uncertainties in the position and momentum is, therefore, λ times h/λ or
h, i.e. ∆p ∆x = h/λ × λ = h.
Thus, we see in accordance with the uncertainty principle that h is the
minimum value for the product of the uncertainties of the position and
the momentum. There is no way of avoiding the uncertainty principle. In
a discussion with my students it was suggested that the uncertainty in
momentum introduced by the momentum of the photon could be avoided
by shooting photons at the particle equally from all sides. This ingenious