Atomic Structure Physics Lesson 41 by Owen Borville 1.18.2026
Atoms are the smallest unit of elements that can combine to form molecules, the smallest unit of compounds. The first direct observation of atoms was with Brownian motion, the analysis of which gave accurate sizes for atoms (10^-10 m on average) and a precise value for Avogadro's number.
Atoms are composed of negatively charged electrons, first proved to exist in cathode-ray-tube experiments, and a positively charged nucleus. All electrons are identical and have a charge-to-mass ratio of qe/me = -1.76 x 10^11 C/kg.
The positive charge in the nuclei is carried by particles called protons, which have a charge-to-mass ratio of qp/mp = 9.57 x 10^7 C/kg.
Mass of electron = me = 9.11 x 10^-31 kg.
Mass of proton = mp = 1.67 x 10^-27 kg.
The planetary model of the atom pictures electrons orbiting the nucleus in the same way that planets orbit the sun.
Bohr's atomic model used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula 1/λ = R[(1/nf^2) - (1/ni^2)], where λ is the wavelength of the emitted EM radiation and R is the Rydberg constant, R = 1.097 x 10^7 m-1. The constants ni and nf are positive integers, and ni must be greater than nf. Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits = ΔE = hf = Ei-Ef, where ΔE is the change in energy between the initial and final orbits and hf is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum (L) = (me)v(rn) = n(h/2π)(n=1, 2, 3,...) where L is the angular momentum, rn is the radius of the nth orbit, and h is Plank's constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is = r(n) = [n^2)/Z](aB) (allowed orbits n = 1, 2, 3...), Z is the atomic number of an element (the number of electrons it has when neutral) and aB is defined to be the Bohr radius = aB = (h^2)/[(4π^2)(me)k(qe^2)] = 0.529 x 10^-10 m.
The energies of hydrogen like atoms = En = -(Z^2)/(n^2)E0 (n=1, 2, 3...), where E0 is the ground state energy = E0 = (2 π^2)(qe^4)m(e)(k^2)/(h^2) = 13.6 eV.
For hydrogen, En = -13.6 eV/n^2(n=1, 2, 3, ...)
The maximum number of electrons in a subshell of a hydrogen atom is N = 4l + 2. The selection rule for atomic transitions in a hydrogen-like atom is Δl = +-1
The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.
X-rays are relatively high-frequency EM radiation and are produced by transitions between inner-shell electron levels, which produce x rays characteristic of the atomic element, or by decelerating electrons. X-rays have many uses, including medical diagnostics and x-ray diffraction.
Fluorescence is an important atomic process where an atom or molecule is excited by absorbing a photon of a given energy and de-excited by emitting a photon of a lower energy. Some states live much longer than others (metastable state). Phosphorescence is the de-excitation of a metastable state. Lasers produce coherent single-wavelength EM radiation by stimulated emission, in which a metastable state is stimulated to decay. Lasing requires a population inversion, in which a majority of the atoms or molecules are in their metastable state.
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. nλ(n) = 2πr(n) (n = 1, 2, 3...), where λn is the electron's de Broglie wavelength.
Because of the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits, but 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 = ΔE = hf = Ei-Ef, where ΔE is the change in energy between the initial and final orbits and 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, Bohn proposed, and must have quantized orbital angular momentum L = m(e)vr(n) = nh/2π(n = 1, 2, 3...) where L is the angular momentum, r(n) is the radius of orbit n, and h is Planck's constant.
The Zeeman effect is a spectra pattern revealing more quantization by the splitting of lines when a magnetic field is applied and is caused by other quantized entities in atoms. Both the magnitude and direction of orbital angular momentum are quantized and the same is true for the magnitude and direction of the intrinsic spin of electrons.
Quantum numbers are used to express the allowed values of quantized entities. The principle quantum number n labels the basic states of a system and is given by n = 1, 2, 3,... The magnitude of angular momentum is = L = √l(l+1)h/2π (l = 0, 1, 2, ..., n-1) where l is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the z-axis is given by Lz = m1h/2π (ml = -l, -l +1, ...,-1, 0, 1, l-1, l),
Lz is the z-component of angular momentum (Lz = mh) and ml is the angular momentum projection quantum number. Similarly, the electron's intrinsic spin angular momentum s is = S = √s(s+1)h/2π (s = 1/2 for electrons) s is defined to be the spin quantum number.
Finally, the direction of the electron's spin along the z axis is = S(z) = ms(h)/2π (ms = -1/2, +1/2), where Sz is the z-component of spin angular momentum and ms is the spin projection quantum number. Spin projection ms = +1/2 is referred to as spin up, whereas ms = -1/2 is called spin down.
The state of a system is completely described by a complete set of quantum numbers: (n, l, m1, ms). The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers, or no two electrons can be in the same state. This exclusion limits the number of electrons in atomic shells and subshells. Each value of n corresponds to a shell, and each value of l corresponds to a subshell. The maximum number of electrons that can be in a subshell is 2 (2l+1). The maximum number of electrons that can be in a shell is 2n^2.
The hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum. The state of an electron in a hydrogen atom is specified by its quantum numbers (n,l,m). In contrast to the Bohr model of the atom, the Schrodinger model makes predictions based on probability statements. The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.
A hydrogen atom has magnetic properties because the motion of the electron acts as a current loop. The energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field because the orbital angular magnetic momentum interacts with the field. The quantum numbers of an electron in a hydrogen atom can be used to calculate the magnitude and direction of the orbital magnetic dipole moment of the atom.
The state of an electron in a hydrogen atom can be expressed in terms of five quantum numbers. The spin angular momentum quantum of an electron is = +1/2. The spin angular momentum projection quantum number is ms = +1/2 or -1/2 (spin up or spin down). The fine and hyperfine structures of the hydrogen spectrum are explained by magnetic interactions within the atom.
(Wolfgang) Pauli's exclusion principle states that no two electrons in an atom can have all the same quantum numbers. The structure of the periodic table of elements can be explained in terms of the total energy, orbital angular momentum, and spin of electrons in an atom. The state of an atom can be expressed by its electron configuration, which describes the orbital shells and subshells that are filled in the atom.
Radiation is absorbed and emitted by atomic energy-level transitions. Quantum numbers can be used to estimate the energy, frequency, and wavelength of photons produced by atomic transitions. Atomic fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon. X-ray photons are produced when a vacancy in an inner shell of an atom is filled by an electron from the outer shell of the atom. The frequency of X-ray radiation is related to the atomic number Z of an atom.
Lasers (laser light) is coherent (monochromatic and "phase linked") light. Laser light is produced by population inversion and subsequent de-excitation of electrons in a material (solid, liquid, or gas). CD and Blu-Ray players use lasers to read digital information stored on discs.
Atoms are the smallest unit of elements that can combine to form molecules, the smallest unit of compounds. The first direct observation of atoms was with Brownian motion, the analysis of which gave accurate sizes for atoms (10^-10 m on average) and a precise value for Avogadro's number.
Atoms are composed of negatively charged electrons, first proved to exist in cathode-ray-tube experiments, and a positively charged nucleus. All electrons are identical and have a charge-to-mass ratio of qe/me = -1.76 x 10^11 C/kg.
The positive charge in the nuclei is carried by particles called protons, which have a charge-to-mass ratio of qp/mp = 9.57 x 10^7 C/kg.
Mass of electron = me = 9.11 x 10^-31 kg.
Mass of proton = mp = 1.67 x 10^-27 kg.
The planetary model of the atom pictures electrons orbiting the nucleus in the same way that planets orbit the sun.
Bohr's atomic model used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula 1/λ = R[(1/nf^2) - (1/ni^2)], where λ is the wavelength of the emitted EM radiation and R is the Rydberg constant, R = 1.097 x 10^7 m-1. The constants ni and nf are positive integers, and ni must be greater than nf. Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits = ΔE = hf = Ei-Ef, where ΔE is the change in energy between the initial and final orbits and hf is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum (L) = (me)v(rn) = n(h/2π)(n=1, 2, 3,...) where L is the angular momentum, rn is the radius of the nth orbit, and h is Plank's constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is = r(n) = [n^2)/Z](aB) (allowed orbits n = 1, 2, 3...), Z is the atomic number of an element (the number of electrons it has when neutral) and aB is defined to be the Bohr radius = aB = (h^2)/[(4π^2)(me)k(qe^2)] = 0.529 x 10^-10 m.
The energies of hydrogen like atoms = En = -(Z^2)/(n^2)E0 (n=1, 2, 3...), where E0 is the ground state energy = E0 = (2 π^2)(qe^4)m(e)(k^2)/(h^2) = 13.6 eV.
For hydrogen, En = -13.6 eV/n^2(n=1, 2, 3, ...)
The maximum number of electrons in a subshell of a hydrogen atom is N = 4l + 2. The selection rule for atomic transitions in a hydrogen-like atom is Δl = +-1
The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.
X-rays are relatively high-frequency EM radiation and are produced by transitions between inner-shell electron levels, which produce x rays characteristic of the atomic element, or by decelerating electrons. X-rays have many uses, including medical diagnostics and x-ray diffraction.
Fluorescence is an important atomic process where an atom or molecule is excited by absorbing a photon of a given energy and de-excited by emitting a photon of a lower energy. Some states live much longer than others (metastable state). Phosphorescence is the de-excitation of a metastable state. Lasers produce coherent single-wavelength EM radiation by stimulated emission, in which a metastable state is stimulated to decay. Lasing requires a population inversion, in which a majority of the atoms or molecules are in their metastable state.
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. nλ(n) = 2πr(n) (n = 1, 2, 3...), where λn is the electron's de Broglie wavelength.
Because of the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits, but 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 = ΔE = hf = Ei-Ef, where ΔE is the change in energy between the initial and final orbits and 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, Bohn proposed, and must have quantized orbital angular momentum L = m(e)vr(n) = nh/2π(n = 1, 2, 3...) where L is the angular momentum, r(n) is the radius of orbit n, and h is Planck's constant.
The Zeeman effect is a spectra pattern revealing more quantization by the splitting of lines when a magnetic field is applied and is caused by other quantized entities in atoms. Both the magnitude and direction of orbital angular momentum are quantized and the same is true for the magnitude and direction of the intrinsic spin of electrons.
Quantum numbers are used to express the allowed values of quantized entities. The principle quantum number n labels the basic states of a system and is given by n = 1, 2, 3,... The magnitude of angular momentum is = L = √l(l+1)h/2π (l = 0, 1, 2, ..., n-1) where l is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the z-axis is given by Lz = m1h/2π (ml = -l, -l +1, ...,-1, 0, 1, l-1, l),
Lz is the z-component of angular momentum (Lz = mh) and ml is the angular momentum projection quantum number. Similarly, the electron's intrinsic spin angular momentum s is = S = √s(s+1)h/2π (s = 1/2 for electrons) s is defined to be the spin quantum number.
Finally, the direction of the electron's spin along the z axis is = S(z) = ms(h)/2π (ms = -1/2, +1/2), where Sz is the z-component of spin angular momentum and ms is the spin projection quantum number. Spin projection ms = +1/2 is referred to as spin up, whereas ms = -1/2 is called spin down.
The state of a system is completely described by a complete set of quantum numbers: (n, l, m1, ms). The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers, or no two electrons can be in the same state. This exclusion limits the number of electrons in atomic shells and subshells. Each value of n corresponds to a shell, and each value of l corresponds to a subshell. The maximum number of electrons that can be in a subshell is 2 (2l+1). The maximum number of electrons that can be in a shell is 2n^2.
The hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum. The state of an electron in a hydrogen atom is specified by its quantum numbers (n,l,m). In contrast to the Bohr model of the atom, the Schrodinger model makes predictions based on probability statements. The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.
A hydrogen atom has magnetic properties because the motion of the electron acts as a current loop. The energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field because the orbital angular magnetic momentum interacts with the field. The quantum numbers of an electron in a hydrogen atom can be used to calculate the magnitude and direction of the orbital magnetic dipole moment of the atom.
The state of an electron in a hydrogen atom can be expressed in terms of five quantum numbers. The spin angular momentum quantum of an electron is = +1/2. The spin angular momentum projection quantum number is ms = +1/2 or -1/2 (spin up or spin down). The fine and hyperfine structures of the hydrogen spectrum are explained by magnetic interactions within the atom.
(Wolfgang) Pauli's exclusion principle states that no two electrons in an atom can have all the same quantum numbers. The structure of the periodic table of elements can be explained in terms of the total energy, orbital angular momentum, and spin of electrons in an atom. The state of an atom can be expressed by its electron configuration, which describes the orbital shells and subshells that are filled in the atom.
Radiation is absorbed and emitted by atomic energy-level transitions. Quantum numbers can be used to estimate the energy, frequency, and wavelength of photons produced by atomic transitions. Atomic fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon. X-ray photons are produced when a vacancy in an inner shell of an atom is filled by an electron from the outer shell of the atom. The frequency of X-ray radiation is related to the atomic number Z of an atom.
Lasers (laser light) is coherent (monochromatic and "phase linked") light. Laser light is produced by population inversion and subsequent de-excitation of electrons in a material (solid, liquid, or gas). CD and Blu-Ray players use lasers to read digital information stored on discs.