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Standing Electron Waves on a Circular Orbit
 
The so-called Bohr model played a crucial role in developing the modern theory of the atomic structure. In 1913, the great Danish physicist Niels Bohr formulated the following three postulates:

1. Each atom has a set of possible energy levels (stable states) . An atom can have an amount of internal energy that is equal to any of these levels, but it cannot have an amount of energy that is intermediate between these levels. In stable states, atoms do not radiate energy.

2. An atom can make a transition from one energy level to another by emitting a quantum of electro-magnetic energy (photon) . The energy of an emitted or absorbed photon is equal energy difference between the levels:
hν = En - Em
where h = 6.63*10-34 J s (Planck's constant), and ν is the photon's frequency.

3. The third postulate assumes that an electrons in an atom revolve in certain circular stable orbits with a quantum angular momentum:
mvrn = n(h/2π)
where m is the electron's mass, v is its speed, and rn is the orbit radius. The integer n is the quantum number.
In 1924, French physicist de Broglie took Bohr?s postulates a step further by addressing them in a theory of his own. According to de Broglie, an electron (or any other microscopic object) with mass m and moving at velocity v should have a wavelength of
λ = h/mv = h/p.
In the case of an electron's motion in the hydrogen atom, Bohr's postulates result in
nλ = 2πrn.
In order for the wave traveling along a circular orbit -- as suggested by de Broglie -- to form a smooth geometric shape, the circumference of the orbit must be an integral number of wavelengths.
For the hydrogen atom
rn = n2r1,
Where r1 = 5?10-11 m is Bohr radius.
Therefore, Bohr's third postulate is an equivalent to a supposition of existence of a standing electron waves on stable circular orbits.

Today, the notion of an electron moving in circular orbits has no real scientific relevance. Quantum physicists now describe the electron using the so-called wave functions, and they have given up the Bohrean notion of the electron trajectories.
 
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