Quantized energy levels - Revision history

Nanderson92: /* The Main Idea */

2025-12-02T04:45:30Z

The Main Idea

← Older revision		Revision as of 00:45, 2 December 2025
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	by NATHAN ANDERSON FALL 2025		by NATHAN ANDERSON FALL 2025

			In this section, we use the Bohr Model of the hydrogen atom: electrons are treated as particles in circular "orbits" with quantized energies. This model is historically important and gives the right energy levels for hydrogen and hydrogen-like ions, but it is not the full modern quantum-mechanical picture (which uses wavefunctions/orbitals instead of literal orbits).

	The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.		The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.

Nanderson92: /* Mathematical Model */

2025-12-02T04:40:01Z

Mathematical Model

← Older revision		Revision as of 00:40, 2 December 2025
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	As we know from the background information, Niels Bohr corrected Rutherford's planetary model by proposing every electron's translational angular momentum is quantized: <math>\|\vec L_{trans,\ nucleus}\|=N\hbar</math>		As we know from the background information, Niels Bohr corrected Rutherford's planetary model by proposing every electron's translational angular momentum is quantized: <math>\|\vec L_{trans,\ nucleus}\|=N\hbar</math>

	Where "N" is the orbit numbers<math>(= 1,2,3...)</math> and <math>\hbar=\frac{h}{2\pi}=1.05\times10^{-34}</math> with "h" ~~plank~~'s constant.		Where "N" is the orbit numbers<math>(= 1,2,3...)</math> and <math>\hbar=\frac{h}{2\pi}=1.05\times10^{-34}</math> with "h" planck's constant.

	Moreover, by the definition of angular momentum, its magnitude is given by <math>\|\vec L\|=\|\vec r\times\vec p\|=rmv </math>, where "r" is the perpendicular distance from the center of the nucleus to the orbit (i.e. the orbital radius) and "mv" is the linear momentum of an individual electron.		Moreover, by the definition of angular momentum, its magnitude is given by <math>\|\vec L\|=\|\vec r\times\vec p\|=rmv </math>, where "r" is the perpendicular distance from the center of the nucleus to the orbit (i.e. the orbital radius) and "mv" is the linear momentum of an individual electron.

Nanderson92: /* The Main Idea */

2025-12-02T04:39:45Z

The Main Idea

← Older revision		Revision as of 00:39, 2 December 2025
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	The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.		The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.

	The various levels of energy and orbital radii associated with an electron are described using its ~~principle~~ quantum number (often denoted as <math>n</math>). A ~~principle~~ quantum number <math>n</math> of 1 indicates that the electron is in the orbit (or 'shell') closest to the nucleus; this state is of the lowest energy level and is referred to as the 'ground state'. A ~~principle~~ quantum number of greater than 1 indicates an electron in a larger orbit and higher energy level; this state is referred to as 'excited'. Less energy is required to free an electron from an excited state.		The various levels of energy and orbital radii associated with an electron are described using its principal quantum number (often denoted as <math>n</math>). A principal quantum number <math>n</math> of 1 indicates that the electron is in the orbit (or 'shell') closest to the nucleus; this state is of the lowest energy level and is referred to as the 'ground state'. A principal quantum number of greater than 1 indicates an electron in a larger orbit and higher energy level; this state is referred to as 'excited'. Less energy is required to free an electron from an excited state.

	The energy levels and radii of orbiting electrons are rigid and not continuous. When an electron transitions to an orbit of greater radius and thus higher energy it does so automatically. At the ground state (n = 1) an electron orbiting Hydrogen has an energy of -13.1 eV and orbits the atom at a radius of 0.0529 nm. At the first excited state (n = 2) the electron has an energy of -3.~~275~~ eV and orbits at a radius of .2116 nm. The electron at no point has an energy between -13.1 eV and -3.275 eV and at no point exists between .0529 nm and .2116 nm, but rather only at those exact values and any change between them is instantaneous. This phenomenon is similar to jumping up and down a flight of stairs. You can only stand on the steps and no where in between. Similarly, the electron can only exist at the exact radii and energy levels given by plugging its ~~principle~~ quantum number <math> n </math> into [http://physicsbook.gatech.edu/Bohr_Model#Angular_Momentum_Is_Quantized the relevant formulas]. In problem sets this number will either be given or easily derived from given information.		The energy levels and radii of orbiting electrons are rigid and not continuous. When an electron transitions to an orbit of greater radius and thus higher energy it does so automatically. At the ground state (n = 1) an electron orbiting Hydrogen has an energy of -13.6 eV and orbits the atom at a radius of 0.0529 nm. At the first excited state (n = 2) the electron has an energy of -3.4 eV and orbits at a radius of .2116 nm. The electron at no point has an energy between -13.1 eV and -3.275 eV and at no point exists between .0529 nm and .2116 nm, but rather only at those exact values and any change between them is instantaneous. This phenomenon is similar to jumping up and down a flight of stairs. You can only stand on the steps and no where in between. Similarly, the electron can only exist at the exact radii and energy levels given by plugging its principal quantum number <math> n </math> into [http://physicsbook.gatech.edu/Bohr_Model#Angular_Momentum_Is_Quantized the relevant formulas]. In problem sets this number will either be given or easily derived from given information.

	[[File:Absorption spectrum.jpg]]		[[File:Absorption spectrum.jpg]]

Nanderson92: /* The Main Idea */

2025-12-02T04:38:12Z

The Main Idea

← Older revision		Revision as of 00:38, 2 December 2025
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	==The Main Idea==		==The Main Idea==

			by NATHAN ANDERSON FALL 2025

	The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.		The negatively charged electrons around an atomic nucleus are held in orbit by attraction to the positively charged protons in the nucleus. Electrons are only permitted to exist in rigidly defined orbits known as a "stationary orbit" with specific radii which correspond to specific energy levels. The lowest energy and smallest radius orbit of one hydrogen electron has the exact same radius and energy level of any other hydrogen electron. This behavior occurs due to the wave-particle duality of electrons and results in a formulaic and regular behavior of energy levels in each stationary orbit. Because the energy levels of each orbit are not a continuous spectrum, and rather exist at discrete values, they are said to be 'quantized'. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.
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	If light is shone through a gas, the gas will absorb the specific wavelengths characteristic of the atoms in the gas. If the light were to be put through a prism of light or a diffraction grating, then there would be absorption lines, or places where the wavelength of light had been absorbed into the gas. This process creates something called an absorption spectrum. Similarly, if this same gas was heated to the right temperature, it would emit the same wavelengths that it absorbed before. Putting this emitted light through a prism or diffraction grating would create an emission spectrum. This is the opposite of an absorption spectrum because it shows the emission lines from the gas instead of the absorption lines. <br/>		If light is shone through a gas, the gas will absorb the specific wavelengths characteristic of the atoms in the gas. If the light were to be put through a prism of light or a diffraction grating, then there would be absorption lines, or places where the wavelength of light had been absorbed into the gas. This process creates something called an absorption spectrum. Similarly, if this same gas was heated to the right temperature, it would emit the same wavelengths that it absorbed before. Putting this emitted light through a prism or diffraction grating would create an emission spectrum. This is the opposite of an absorption spectrum because it shows the emission lines from the gas instead of the absorption lines. <br/>


	==Mathematical Model==		==Mathematical Model==

Jamesclw: /* Mathematical Model */

2024-12-03T20:20:50Z

Mathematical Model

← Older revision		Revision as of 16:20, 3 December 2024
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	Formula Derivation: <math> E_{total} = K+U_{electromagnetic}=\frac{mv^2}{2}+(-\frac{\frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{me^2}{\hbar}}{N^2})=-\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>		Formula Derivation: <math> E_{total} = K+U_{electromagnetic}=\frac{mv^2}{2}+(-\frac{\frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{me^2}{\hbar}}{N^2})=-\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>


	<br />2) Radius of orbit: <math> r_{n} = a_{0}n^2 </math>		<br />2) Radius of orbit: <math> r_{n} = a_{0}n^2 </math>

Jamesclw: /* Mathematical Model */

2024-12-03T20:20:22Z

Mathematical Model

← Older revision		Revision as of 16:20, 3 December 2024
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	* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />		* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />

	Formula Derivation: <math> E_{total} = K+U_{electromagnetic}=\frac{mv^2}{2}+\frac{\frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{me^2}{\hbar}}{N^2}=-\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>		Formula Derivation: <math> E_{total} = K+U_{electromagnetic}=\frac{mv^2}{2}+(-\frac{\frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{me^2}{\hbar}}{N^2})=-\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>

Jamesclw: /* Mathematical Model */

2024-12-03T20:19:49Z

Mathematical Model

← Older revision		Revision as of 16:19, 3 December 2024
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	* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />		* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />

	Formula Derivation: <math> E_{total} = -\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>		Formula Derivation: <math> E_{total} = K+U_{electromagnetic}=\frac{mv^2}{2}+\frac{\frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{me^2}{\hbar}}{N^2}=-\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>

Jamesclw: /* Mathematical Model */

2024-12-03T20:15:43Z

Mathematical Model

← Older revision		Revision as of 16:15, 3 December 2024
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	* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />		* Where <math>n</math> is the principle quantum number. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.<br />

	~~This formula is from~~: <math> E_{total} = -\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>		Formula Derivation: <math> E_{total} = -\frac{1}{2}\frac{me^4}{(4\pi\epsilon_0)^2\hbar^2}\frac{1}{N^2} </math>

Jamesclw: /* Challenging */

2024-12-03T20:11:57Z

Challenging

← Older revision		Revision as of 16:11, 3 December 2024
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	5) ∆<math>E_{n}</math> = <math> 12.75eV </math>		5) ∆<math>E_{n}</math> = <math> 12.75eV </math>

	===~~Difficult~~===		===Challenging===

	What is the energy of a Hydrogen electron in an orbit of radius .4761 nm?		What is the energy of a Hydrogen electron in an orbit of radius .4761 nm?

Jamesclw: /* Medium */

2024-12-03T20:11:44Z

Medium

← Older revision		Revision as of 16:11, 3 December 2024
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	print(((2kqq)/(m_er))**0.5)		print(((2kqq)/(m_er))**0.5)

	===~~Middling~~===		===Medium===

	What is the difference in energy between an electron orbiting a hydrogen atom in the <math>n</math> = 1 orbit, or ground state, and an electron orbiting a different hydrogen atom in the third excited orbit?		What is the difference in energy between an electron orbiting a hydrogen atom in the <math>n</math> = 1 orbit, or ground state, and an electron orbiting a different hydrogen atom in the third excited orbit?