Bohr was able to explain the discrete spectral lines emitted by the hydrogen atom by forcing the electrons into a limited number of permitted orbits (Figure 94a). However, like Planck before him, he did this without having a physical justification.
Figure 94 De Broglie used his proposed matter waves to explain why Bohr’s atomic orbits would be quantized. (a) Bohr explained the discrete spectral lines of hydrogen by enforcing a limited number of orbits. However, he did this without having a physical justification. (b) De Broglie proposed that, like in a guitar string fastened at the ends to rigid supports, allowed orbits would be those where a complete number of electron waves would fit on the orbit. (c) A higher-energy orbit would thus fit a larger number of complete vibrations than a lower-energy orbit.
The first theoretical success of de Broglie’s matter waves came when he used them to explain why certain electron orbits are allowed, and why others are not. He proposed that the allowed orbits for an electron are those that support a standing wave of specific wavelength, energy, and frequency (i.e., Bohr’s energy levels); much like a guitar string sets up a standing wave when plucked.
As shown in Figure 94b, de Broglie proposed that if one were to straighten an orbit into a string and fix the ends to rigid supports, it could be set to vibrate. However, since the ends are fixed, the only vibrations that it can support are those where full wavelengths fit between the ends. De Broglie proposed that these vibrations would correspond to his matter waves, so a higher frequency vibration would correspond to an electron at a higher energy state. As shown in Figure 94c, de Broglie’s view of Bohr’s atom consisted of electron waves of different wavelengths. An electron wave would transition between these wavelengths (each at one of Bohr’s energy levels), giving off or absorbing photons with an energy hf equal to the difference in energy ΔE between the electron waves.
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