- #1
Physicsiscool
- 10
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What concepts lead to the quantination of radii in the hydrogen atom, for example: r = r(o) * n squared?
Quantizing Radii in Hydrogen Atom: Concepts & Examples
- Thread starter Physicsiscool
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In summary, the quantization of radii in the hydrogen atom is determined by the discrete spectra of the energy and angular momentum operators, which are chosen based on satisfying the classical correspondence limit and the canonical quantum mechanical relations. This quantization arises from the assumptions of electrical attraction between the electron and proton and the requirement that the path length be a multiple of the de Broglie wavelength. This concept is much deeper than just considering de Broglie waves and trajectories and is not typically taught at a high school level.
- #2
masudr
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The energy and angular momentum operators have discrete spectra when applied to the system of a hydrogen atom. And angular momentum being momentum * radius means that the radius is quantised.
To understand why the operators are quantised, you have to go all the way back to how we choose quantum operators, and that's down to satisfying the classical correspondence limit (in the limit as masses and momentums become large, results predicted by QM tends to results predicted by classical mechanics) as well as satisfying the canonical quantum mechanical relations (pq - qp = -ih etc).
To understand what these operators work on, you should be familiar with the Hilbert space. You should also understand Schrodinger's equation - that's what we use to derive radii levels.
To understand why the operators are quantised, you have to go all the way back to how we choose quantum operators, and that's down to satisfying the classical correspondence limit (in the limit as masses and momentums become large, results predicted by QM tends to results predicted by classical mechanics) as well as satisfying the canonical quantum mechanical relations (pq - qp = -ih etc).
To understand what these operators work on, you should be familiar with the Hilbert space. You should also understand Schrodinger's equation - that's what we use to derive radii levels.
- #3
Leo32
- 31
- 1
I read a text which showed this quantisation to come foreward from 2 assumptions:
- the electrical attraction between electron/proton keeping the electron in a circular path around the proton
- the fact that the length of the path needs to be a multiple of the brogly wavelength
It's too early in the morning at work to dash out the formula's from my head, but it should give you an idea.
Greetz,
Leo
- the electrical attraction between electron/proton keeping the electron in a circular path around the proton
- the fact that the length of the path needs to be a multiple of the brogly wavelength
It's too early in the morning at work to dash out the formula's from my head, but it should give you an idea.
Greetz,
Leo
- #4
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The ideas of quantization are much more deeper than to picture all sorts of de Broglie waves and how they fit into trajectories,concepts which have nothing to do with the QM.This description is basically taught at high-school level and it's for the mass,not for the ones who are interested in going under he surface.I remember my 12-th grade manual,it was so stupid,when talking Bohr Hydrogen atom,it mentioned de Broglie's quantizing condition...I threw it away.Thankfully me and the teacher were lot smarter than the authors... 
- #5
Physicsiscool
- 10
- 0
This has been very helpful. Thank you!
FAQ: Quantizing Radii in Hydrogen Atom: Concepts & Examples
What is quantization in the context of the hydrogen atom?
Quantization in the context of the hydrogen atom refers to the discrete energy levels that an electron can occupy in the atom's orbitals. These energy levels are determined by the electron's distance from the nucleus and are represented by the principal quantum number, n.
How do you calculate the quantized radii in the hydrogen atom?
The quantized radii in the hydrogen atom can be calculated using the Bohr model, which states that the radius of an electron's orbit is proportional to n², where n is the principal quantum number. The formula for calculating the quantized radius is r = n²a₀, where a₀ is the Bohr radius (0.529 Å).
What are some examples of quantized radii in the hydrogen atom?
Some examples of quantized radii in the hydrogen atom include the first energy level (n=1) with a radius of 0.529 Å, the second energy level (n=2) with a radius of 2.12 Å, and the third energy level (n=3) with a radius of 4.77 Å. These values can be calculated using the formula r = n²a₀.
How does quantization in the hydrogen atom relate to the emission spectrum?
The quantization of energy levels in the hydrogen atom is directly related to the emission spectrum of the atom. When an electron transitions from a higher energy level to a lower one, it emits a photon of light with a specific wavelength corresponding to the difference in energy between the two levels. This results in the distinct lines observed in the hydrogen emission spectrum.
Why is quantization important in understanding the behavior of the hydrogen atom?
Quantization is important in understanding the behavior of the hydrogen atom because it explains why the atom's emission spectrum consists of distinct lines rather than a continuous spectrum. It also helps to explain the stability of the atom, as the quantized energy levels prevent the electron from spiraling into the nucleus. Additionally, the concept of quantization is a fundamental principle in quantum mechanics and is necessary for understanding the behavior of all atoms and particles.