Page 233 - Instant notes

Quantization of energy and particle-wave duality 219

where h is Planck’s constant. The de Broglie relationship implies that faster moving
particles have shorter wavelengths and that, for a given velocity, heavier particles have
shorter wavelengths than lighter ones.
The relationship is confirmed by electron diffraction experiments in which the
wavelength of the associated diffraction pattern matches the calculated momentum of the
electrons. For example, the wavelength of an electron accelerated from rest by a voltage
υ2,
of 2.0 kV is calculated as follows. The kinetic energy, ½m e attained by the electron is
equal to eV, where V is the acceleration voltage and e the charge on the electron
(1.602×10 −19 C). Since momentum, p=mυ:

Combining with de Broglie’s equation gives:

and a numerical value of λ=2.7×10 −11 m in this example.
A wavelength of 27 pm is comparable to the spacing between molecules in crystalline
solids and explains how a beam of electrons can produce a diffraction pattern from a
crystal. For comparison, the wavelength of a cricket ball of mass 0.1 kg travelling at 10 m
−1
s is 6.63×10 −34 m. This wavelength is many orders of magnitude smaller than a sub-
atomic particle and shows why quantum mechanical phenomena are not important for
macroscale objects.