The host of experimental data and the theoretical ideas of Planck and Einstein conclusively established that radiation has a DUAL behavior. Namely it behaves both as
a wave in interference and diffraction and as a particle in photoelectric and compton effect. Inspired by this and guided by the close correspondence of the Fermats principle of optical path in geometrical optics ( particle like behavior) and the
Least Action principle in Classical Analytical Mechanics, de Broglie postulated that
Matter must also exhibit wave like behaviour.
Since for radiation E=hν and p=h/λ de Broglie postulated that the wavelength of matter waves was also given as λDB=h/p where p is the momentum of the particle. A quick calculation with standard values show that for macroscopic systems and velocities λDB is negligibly small, for a 1.0 kg mass moving at 10 m/s the λDB =6.6 x 10 -35 m which is too small to be detectable. But for a 100 eV energy electron it was of the order of 1 Angstrom.
From Optics we know that when λ >> a where a is the aperture dimension there is no difraction and geomterical optics holds. For diffraction λ ~ a.
So obviously to have observable diffraction effects λDB ~ a which is 1 angstorm. Such gratings are offered by the ordered periodic arrays of atomic layers in a crystal. The inter layer spacing is of the order of 1 angstorm.
So an electron beam scattered from such a crystal should exhibit diffraction.
This experiment was first performed by Davisson and Germer
the results of the experiment are summarized here
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/davger2.html#c1
The results clearly showed that the wave particle duality was valid also for matter just as in radiation and obeyed the same formula for both the momentum and energy namely E=hν and p=h/λ
Bohr's Complementarity Principle : This principle asserts that the wave and the particle behaviours for both matter and radiation are complementary. In a single experiment either the wave nature or the particle nature is manifest but never together.
So radiation and matter are neither simply waves nor simply particles. They are more general objects. In advanced applications the view of radiation and matter that emerges is a general description in terms of quantized fields which displays both particle (quanta) and wave ( field) properties. Particles are described as the field quanta
The link between the particle and the wave picture is provided by an interpretation of the wave particle duality based on probability. In the wave picture the Intensity I ~ Eav2 where Eav is the average electric field ( amplitude) over 1 cycle. In the photon picture I ~ Navhν where Nav is the average number of photos crossing unit area/unit time perpendicular to the propagation vector of the EM wave. In EM theory Eav2 is proportional to the energy density. Einstein interpreted this as the average photon density which owing to the statistical nature of emission was related to the probability measure for a photon to cross unit area per unit time. So I~Eav2 ~Navhν .
Born borrowed this probability interpretation later to apply to the de Broglie's matter waves also. Since probability and intensity had a simmilar behaviour and the intensity is related to the square of an amplitude, one thinks of a probability wave
amplitude whose square is related to the probability. This wave is a probability wave
whose amplitude squared is related to the probability. This probability wave should satisfy a linear wave equation simmilar to the wave equation satisfied by a standard
EM wave for the superposition principle to be valid.
a wave in interference and diffraction and as a particle in photoelectric and compton effect. Inspired by this and guided by the close correspondence of the Fermats principle of optical path in geometrical optics ( particle like behavior) and the
Least Action principle in Classical Analytical Mechanics, de Broglie postulated that
Matter must also exhibit wave like behaviour.
Since for radiation E=hν and p=h/λ de Broglie postulated that the wavelength of matter waves was also given as λDB=h/p where p is the momentum of the particle. A quick calculation with standard values show that for macroscopic systems and velocities λDB is negligibly small, for a 1.0 kg mass moving at 10 m/s the λDB =6.6 x 10 -35 m which is too small to be detectable. But for a 100 eV energy electron it was of the order of 1 Angstrom.
From Optics we know that when λ >> a where a is the aperture dimension there is no difraction and geomterical optics holds. For diffraction λ ~ a.
So obviously to have observable diffraction effects λDB ~ a which is 1 angstorm. Such gratings are offered by the ordered periodic arrays of atomic layers in a crystal. The inter layer spacing is of the order of 1 angstorm.
So an electron beam scattered from such a crystal should exhibit diffraction.
This experiment was first performed by Davisson and Germer
the results of the experiment are summarized here
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/davger2.html#c1
The results clearly showed that the wave particle duality was valid also for matter just as in radiation and obeyed the same formula for both the momentum and energy namely E=hν and p=h/λ
Bohr's Complementarity Principle : This principle asserts that the wave and the particle behaviours for both matter and radiation are complementary. In a single experiment either the wave nature or the particle nature is manifest but never together.
So radiation and matter are neither simply waves nor simply particles. They are more general objects. In advanced applications the view of radiation and matter that emerges is a general description in terms of quantized fields which displays both particle (quanta) and wave ( field) properties. Particles are described as the field quanta
The link between the particle and the wave picture is provided by an interpretation of the wave particle duality based on probability. In the wave picture the Intensity I ~ Eav2 where Eav is the average electric field ( amplitude) over 1 cycle. In the photon picture I ~ Navhν where Nav is the average number of photos crossing unit area/unit time perpendicular to the propagation vector of the EM wave. In EM theory Eav2 is proportional to the energy density. Einstein interpreted this as the average photon density which owing to the statistical nature of emission was related to the probability measure for a photon to cross unit area per unit time. So I~Eav2 ~Navhν .
Born borrowed this probability interpretation later to apply to the de Broglie's matter waves also. Since probability and intensity had a simmilar behaviour and the intensity is related to the square of an amplitude, one thinks of a probability wave
amplitude whose square is related to the probability. This wave is a probability wave
whose amplitude squared is related to the probability. This probability wave should satisfy a linear wave equation simmilar to the wave equation satisfied by a standard
EM wave for the superposition principle to be valid.
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