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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.

The figure shows two concentric circular orbits with radius r three and r four. Two curved paths representing electron waves are shown around the two circular orbits.
The third and fourth allowed circular orbits have three and four wavelengths, respectively, in their circumferences.

Because of the wave character of matter, the idea of well-defined orbits gives way to a model in which there is a cloud of probability, consistent with Heisenberg’s uncertainty principle. [link] shows how this applies to the ground state of hydrogen. If you try to follow the electron in some well-defined orbit using a probe that has a small enough wavelength to get some details, you will instead knock the electron out of its orbit. Each measurement of the electron’s position will find it to be in a definite location somewhere near the nucleus. Repeated measurements reveal a cloud of probability like that in the figure, with each speck the location determined by a single measurement. There is not a well-defined, circular-orbit type of distribution. Nature again proves to be different on a small scale than on a macroscopic scale.

A hydrogen atom is shown with its nucleus and most probable distance for the electron. N equals one; l equals zero; m sub l equals zero. R sub one equals a sub B, most probable distance for an electron.
The ground state of a hydrogen atom has a probability cloud describing the position of its electron. The probability of finding the electron is proportional to the darkness of the cloud. The electron can be closer or farther than the Bohr radius, but it is very unlikely to be a great distance from the nucleus.

There are many examples in which the wave nature of matter causes quantization in bound systems such as the atom. Whenever a particle is confined or bound to a small space, its allowed wavelengths are those which fit into that space. For example, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. This is true in blackbody radiators (atoms and molecules) as well as in atomic and molecular spectra. Various atoms and molecules will have different sets of electron orbits, depending on the size and complexity of the system. When a system is large, such as a grain of sand, the tiny particle waves in it can fit in so many ways that it becomes impossible to see that the allowed states are discrete. Thus the correspondence principle is satisfied. As systems become large, they gradually look less grainy, and quantization becomes less evident. Unbound systems (small or not), such as an electron freed from an atom, do not have quantized energies, since their wavelengths are not constrained to fit in a certain volume.

Phet explorations: quantum wave interference

When do photons, electrons, and atoms behave like particles and when do they behave like waves? Watch waves spread out and interfere as they pass through a double slit, then get detected on a screen as tiny dots. Use quantum detectors to explore how measurements change the waves and the patterns they produce on the screen.

Quantum Wave Interference

Test prep for ap courses

The figure shows two graphs representing two particles (Particle X and Particle Y). The vertical y-axis is labeled Amplitude with a point A subzero indicated near the top of the displayed axis. The horizontal x-axis is labeled x with a point labeled x subzero. There is a wave on each graph centered at x subzero. Both waves start with from the left with a small amplitude. The wave for particle X is spread out in the x-direction and increases in amplitude until A subzero is reached and then gradually tapers off in a mirror image of the beginning portion of the wave. Particle Y’s graph shows a wave that is much less wide in the x-direction that quickly builds past the A subzero amplitude and then quickly decreases.

This figure shows graphical representations of the wave functions of two particles, X and Y , that are moving in the positive x -direction. The maximum amplitude of particle X ’s wave function is A 0 . Which particle has a greater probability of being located at position x 0 at this instant, and why?

  1. Particle X , because the wave function of particle X spends more time passing through x 0 than the wave function of particle Y .
  2. Particle X , because the wave function of particle X has a longer wavelength than the wave function of particle Y .
  3. Particle Y , because the wave function of particle Y is narrower than the wave function of particle X .
  4. Particle Y , because the wave function of particle Y has a greater amplitude near x 0 than the wave function of particle X .

(d)

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In [link] , explain qualitatively the difference in the wave functions of particle X and particle Y . Which particle is more likely to be found at a larger distance from the coordinate x 0 and why? Which particle is more likely be found exactly at x 0 and why?

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For an electron with a de Broglie wavelength λ , which of the following orbital circumferences within the atom would be disallowed? Select two answers.

  1. 0.5 λ
  2. λ
  3. 1.5 λ
  4. 2 λ

(a), (c)

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We have discovered that an electron’s orbit must contain an integer number of de Broglie wavelengths. Explain why, under ordinary conditions, this makes it impossible for electrons to spiral in to merge with the positively charged nucleus.

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Section summary

  • Quantization of orbital energy is caused by the wave nature of matter. Allowed orbits in atoms occur for constructive interference of electrons in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that 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 size 12{λ rSub { size 8{n} } } {} is the electron’s de Broglie wavelength.
  • Owing to the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits; rather, there are clouds of probability.
  • Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
    Δ E = hf = E i E f , size 12{ΔE= ital "hf"=E rSub { size 8{i} } - E rSub { size 8{f} } } {}
    where Δ E size 12{ΔE} {} is the change in energy between the initial and final orbits and hf size 12{ ital "hf"} {} is the energy of an absorbed or emitted photon.
  • It is useful to plot orbit energies on a vertical graph called an energy-level diagram.
  • The allowed orbits are circular, Bohr proposed, and must have quantized orbital angular momentum given by
    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 ),} {}
    where L size 12{L} {} is the angular momentum, r n size 12{r rSub { size 8{n} } } {} is the radius of orbit n size 12{n rSup { size 8{"th"} } } {} , and h size 12{h} {} is Planck’s constant.

Conceptual questions

How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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