De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation

Of all the postulates made by Bohr in his model of the atom, the second postulate is one of the most puzzling. It states that the angular momentum of the electron orbiting around the nucleus is quantised.

The question is why the angular momentum should have only those values that are integral multiples of h/2π. Louis de Broglie explained this puzzle in 1923, ten years after Bohr proposed his model.

Material particles, such as electrons, also have a wave nature. This idea was later verified experimentally for electrons by C. J. Davisson and L. H. Germer in 1927.

De Broglie argued that the electron in its circular orbit, as proposed by Bohr, must be regarded as a particle wave. In analogy with waves travelling on a string, particle waves can also produce standing waves under resonant conditions.

When a string is plucked, many wavelengths are produced. However, only those wavelengths survive which have nodes at the ends and form standing waves in the string.

This means that standing waves are formed when the total distance travelled by a wave down the string and back is equal to one wavelength, two wavelengths, or any integral number of wavelengths. Waves with other wavelengths interfere with themselves upon reflection, and their amplitudes quickly fall to zero.

For an electron moving in the n-th circular orbit of radius r_n, the total distance is the circumference of the orbit.

This means that the circumference of the orbit must contain an integral number of de Broglie wavelengths.

From de Broglie’s relation,

Where p is the magnitude of the electron’s momentum. If the speed of the electron is much less than the speed of light, then the momentum is mv_n​. Therefore,

Using this in the condition

we get

or,

This is the quantum condition proposed by Bohr for the electron's angular momentum.

This relation is the basis for explaining the discrete orbits and energy levels in a hydrogen atom. Thus, de Broglie’s hypothesis provided an explanation for Bohr’s second postulate for the quantisation of angular momentum of the orbiting electron.

The quantised electron orbits and energy states are due to the wave nature of the electron, and only resonant standing waves can persist.

• Bohr’s Model and Its Scope

Bohr’s model uses a classical trajectory picture in which the electron moves around the nucleus like a planet. It correctly predicts the gross features of hydrogenic atoms, especially the frequencies of the radiation emitted or selectively absorbed.

However, the model has several limitations.

• The Bohr model is applicable only to hydrogenic atoms.
• It cannot be extended even to two-electron atoms such as helium.
• Atoms with more than one electron cannot be explained successfully on the lines of Bohr’s model.
• Each electron in a multi-electron atom interacts not only with the positively charged nucleus but also with all other electrons.
• The Bohr model includes the electrical force between the nucleus and the electron, but it does not include the electrical forces between electrons.
• Although the Bohr model correctly predicts the frequencies of light emitted by hydrogenic atoms, it cannot explain the relative intensities of spectral lines.
• In the emission spectrum of hydrogen, some visible frequencies are weak while others are strong, and Bohr’s model cannot account for these intensity variations.