12-12-2012, 11:43 AM
Experimental Evidence; Wave-Particle Duality
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The Need for Quantum Mechanics
Towards the beginning of the century experimental evidence started to emerge that the
classical theory did not give an accurate description of nature when dealing with phenomena
on the atomic level. Among the experiments that demonstrated the inadequacy of the
classical theory we mention the following:
Blackbody radiation: The distribution of emitted heat by a blackbody (i.e. the hypothetical
situation of an idealised body that absorbs all radiation that falls onto it1) is
a universal function E(λ, T), depending on temperature T and wavelength λ = c/ν,
ν being the frequency. The energy density in the cavity u(ν, T) = 4E(λ, T)/ν2
follows experimentally Wien’s law (1894)
The Bohr Atom:
The planetary model of the atom (consisting of a heavy positively
charged nucleus surrounded by negatively charged electrons moving in orbits around
the nucleus) was proposed in 1908 by E. Rutherford on the basis of experiments
with α-particles carried out by H.W. Geiger and E. Marsden. However, by general
principles this model is not in accordance with the classical theory. In fact, since an
accelerating charge emits electromagnetic radiation, the electrons will constantly lose
energy thereby spiralling towards the nucleus and the atom will eventually collapse.
Thus, the classical theory cannot explain the stability of the atom.
The Compton effect:
Experiments by A.H. Compton demonstrated that the scattering
of X-ray radiation when sent through thin metallic foils is not in accordance with the
classical theory. Whereas the classical theory would predict a variation of intensity
of the scattered waves according to (1+cos2 θ), where θ is the scattering angle, and
independent of the wavelength, the experiments showed that the intensity would
exhibit a distinct peak with wavelength shifted from the orignal wavelength which
would depend on the scattering angle proportional to (1 − cos θ). An explanation
of this phenomenon could be given in terms of a collision process of particles rather
than by scattering of waves on atoms according to the classical theory. This led
to the assumption that in certain circumstances monochromatic radiation would
behave like a beam of particles, called photons, of mass zero and energy given by
E = hν, where ν is the frequency of the wave. These particles, following relativistic
kinematics, would have a momentum equal to p = hν/c moving with the speed
of light c. Thus, the Compton effect seemed to indicate the dual wave/particle
behaviour of radiation.
Quantum Mechanics and Wave Packets
When we talk about quantum mechanics we mean the theory that was developed in the
late 1920’s, early 1930’s, i.e. the quantum theory of nonrelativistic particles. In this theory
we usually don’t include the quantum theory of electromagnetic radiation which would
necessitate the incorporation of relativistic effects (since photons move at the speed of
light). Thus, in quantum mechanics we still do not treat radiation and matter on an equal
footing, which is not entirely satisfactory. The theory is, therefore, not yet fully applicable
and has to be expanded in a more extensive theory. The latter theory is usually called
quantum theory of radiation, or quantum field theory, and it came to full bloom well after
the second world war. In this module we can only deal with the more restrictive theory
which is the quantum mechanics of particles 4.
Our first aim is to arrive at a wave description of particles. In order to do that we
have to develop some rough first ideas. What we need is a description that captures
both aspects of a particle: its “particle”-nature (i.e. its behaviour as a localised object in
space) and its “wave”-nature (i.e. the possibility that it expands throughout space and
behaves as wave trains). This leads us to the notion of a wave packet: a superposition
of waves whose wave envelope is more or less localised depending on the frequencies and
wave lengths of the wave components that make up the wave packet.