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  • Explain Bohr’s model of atom.
  • Define and describe quantization of angular momentum.
  • Calculate the angular momentum for an orbit of atom.
  • Define and describe the wave-like properties of matter.

After visiting some of the applications of different aspects of atomic physics, we now return to the basic theory that was built upon Bohr’s atom. Einstein once said it was important to keep asking the questions we eventually teach children not to ask. Why is angular momentum quantized? You already know the answer. Electrons have wave-like properties, as de Broglie later proposed. They can exist only where they interfere constructively, and only certain orbits meet proper conditions, as we shall see in the next module.

Following Bohr’s initial work on the hydrogen atom, a decade was to pass before de Broglie proposed that matter has wave properties. The wave-like properties of matter were subsequently confirmed by observations of electron interference when scattered from crystals. Electrons can exist only in locations where they interfere constructively. How does this affect electrons in atomic orbits? When an electron is bound to an atom, its wavelength must fit into a small space, something like a standing wave on a string. (See [link] .) Allowed orbits are those orbits in which an electron constructively interferes with itself. Not all orbits produce constructive interference. Thus only certain orbits are allowed—the orbits are quantized.

Figure a shows a string tied between two fixed supports. The string is being vibrated, which generates waves on the string. Figure b shows a circular orbit of radius r and a triangular shaped wave representing an electron. The condition for constructive interference and an allowed orbit given as two pi r is equal to n times lambda where n is an integer. Figure c shows a circular orbit of radius r prime and an irregular shaped wave representing an electron. The condition for destructive interference and a forbidden orbit is given as two pi r prime is not equal to n times lambda prime where n is an integer.
(a) Waves on a string have a wavelength related to the length of the string, allowing them to interfere constructively. (b) If we imagine the string bent into a closed circle, we get a rough idea of how electrons in circular orbits can interfere constructively. (c) If the wavelength does not fit into the circumference, the electron interferes destructively; it cannot exist in such an orbit.

For a circular orbit, constructive interference occurs when the electron’s wavelength fits neatly into the circumference, so that wave crests always align with crests and wave troughs align with troughs, as shown in [link] (b). More precisely, when an integral multiple of the electron’s wavelength equals the circumference of the orbit, constructive interference is obtained. In equation form, the condition for constructive interference and an allowed electron orbit is

n = 2 πr n size 12{nλ rSub { size 8{n} } =2πr rSub { size 8{n} } } {} n = 1, 2, 3 ... , size 12{ left (n=1, 2, 3 "." "." "." right )} {}

where λ n is the electron’s wavelength and r n is the radius of that circular orbit. The de Broglie wavelength is λ = h / p = h / mv , and so here λ = h / m e v . Substituting this into the previous condition for constructive interference produces an interesting result:

nh m e v = r n . size 12{ { { ital "nh"} over {m rSub { size 8{e} } v} } =2πr rSub { size 8{n} } } {}

Rearranging terms, and noting that L = mvr size 12{L= ital "mvr"} {} for a circular orbit, we obtain the quantization of angular momentum as the condition for allowed orbits:

L = m e vr n = n h size 12{L=m rSub { size 8{e} } ital "vr" rSub { size 8{n} } =n { {h} over {2π} } } {} n = 1, 2, 3 ... . size 12{ left (n=1, 2, 3 "." "." "." right )} {}

This is what Bohr was forced to hypothesize as the rule for allowed orbits, as stated earlier. We now realize that it is the condition for constructive interference of an electron in a circular orbit. [link] illustrates this for n = 3 size 12{n=3} {} and n = 4. size 12{n=3} {}

Waves and quantization

The wave nature of matter is responsible for the quantization of energy levels in bound systems. Only those states where matter interferes constructively exist, or are “allowed.” Since there is a lowest orbit where this is possible in an atom, the electron cannot spiral into the nucleus. It cannot exist closer to or inside the nucleus. The wave nature of matter is what prevents matter from collapsing and gives atoms their sizes.

Questions & Answers

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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