Information about Orbital Angular Momentum
The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers (the principal quantum number, following Spectroscopic notation, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) which describe the unique quantum state of an electron and is designated by the letter l.
An atomic electron's angular momentum, L, which is related to its quantum number
is described by the following equation:
where
is the reduced Planck's constant, also called Dirac's constant, 
is the orbital angular momentum operator and 
is the wavefunction of the electron. While many introductory text books on quantum mechanics will refer to L by itself, L has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is best to simply use the quantum number 
.
The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.
This behavior manifests itself as the "shape" of the orbital.
Electron shells have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the atomic orbital.
Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials. The various orbitals relating to different values of l are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letters, as follows:
A mnemonic for the order of the "shells" is some poor dumb fool. Another mnemonic for the order of the "shells" is silly professors dance funny. The letters after the F subshell just follow F in alphabetical order.
Each of the different angular momentum states can take 2(2l+1) electrons. This is because the third quantum number ml (which can be thought of loosely as the quantised projection of the angular momentum vector on the z-axis) runs from −l to l in integer units, and so there are 2l+1 possible states. Each distinct nlml orbital can be occupied by two electrons with opposing spins (given by the quantum number ms), giving 2(2l+1) electrons overall. Orbitals with higher l than given in the table are perfectly permissible, but these values cover all atoms so far discovered.
For a given value of the principal quantum number, n, the possible values of l range from 0 to n−1; therefore, the n=1 shell only possesses an s subshell and can only take 2 electrons, the n=2 shell possesses an s and a p subshell and can take 8 electrons overall, the n=3 shell possesses s, p and d subshells and has a maximum of 18 electrons, and so on (generally speaking, the maximum number of electrons in the nth energy level is 2n2).
The angular momentum quantum number, l, governs the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and , which has zero magnitude. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number l takes the value of zero. In a p orbital, one node traverses the nucleus and therefore l has the value 1.
Depending on the value of n, the principal quantum number, there is an angular momentum quantum number l and the following series. The wavelengths listed are for a hydrogen atom:
which is the sum of two individual quantized angular momenta 
and 
,
the quantum number
associated with its magnitude can range from 
to 
in integer steps
where 
and 
are quantum numbers corresponding to the magnitudes of the individual angular momenta.
stand for 
, 
and 
.
The quantum numbers describing the system, which are constant over time, are now j and
, defined through the action of J on the wavefunction 
So that j is related to the norm of the total angular momentum and to its projection along a specified axis.
As with any angular momentum in quantum mechanics, the projection of J along other axes cannot be co-defined with, because they do not commute.
Furthermore, the eigenvectors of j, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of l, ml and ms.
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Derivation
There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, l, ml, and s specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation as shown below.An atomic electron's angular momentum, L, which is related to its quantum number
is described by the following equation:
where
is the reduced Planck's constant, also called Dirac's constant, is the orbital angular momentum operator and is the wavefunction of the electron. While many introductory text books on quantum mechanics will refer to L by itself, L has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is best to simply use the quantum number .
The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.
This behavior manifests itself as the "shape" of the orbital.
)
The atomic orbital wavefunctions of a hydrogen atom. The principal quantum number is at the right of each row and the azimuthal quantum number is denoted by letter at top of each column.
Electron shells have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the atomic orbital.
Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials. The various orbitals relating to different values of l are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letters, as follows:
 |
Letter | Max electrons | Shape | Name |
|---|---|---|---|---|
| 0 | s | 2 | sphere | sharp |
| 1 | p | 6 | two dumbbells | principal |
| 2 | d | 10 | four dumbbells | diffuse |
| 3 | f | 14 | eight dumbbells | fundamental |
| 4 | g | 18 | ||
| 5 | h | 22 | | 6 || i || 26 || || |
A mnemonic for the order of the "shells" is some poor dumb fool. Another mnemonic for the order of the "shells" is silly professors dance funny. The letters after the F subshell just follow F in alphabetical order.
Each of the different angular momentum states can take 2(2l+1) electrons. This is because the third quantum number ml (which can be thought of loosely as the quantised projection of the angular momentum vector on the z-axis) runs from −l to l in integer units, and so there are 2l+1 possible states. Each distinct nlml orbital can be occupied by two electrons with opposing spins (given by the quantum number ms), giving 2(2l+1) electrons overall. Orbitals with higher l than given in the table are perfectly permissible, but these values cover all atoms so far discovered.
For a given value of the principal quantum number, n, the possible values of l range from 0 to n−1; therefore, the n=1 shell only possesses an s subshell and can only take 2 electrons, the n=2 shell possesses an s and a p subshell and can take 8 electrons overall, the n=3 shell possesses s, p and d subshells and has a maximum of 18 electrons, and so on (generally speaking, the maximum number of electrons in the nth energy level is 2n2).
The angular momentum quantum number, l, governs the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and , which has zero magnitude. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number l takes the value of zero. In a p orbital, one node traverses the nucleus and therefore l has the value 1.
Depending on the value of n, the principal quantum number, there is an angular momentum quantum number l and the following series. The wavelengths listed are for a hydrogen atom:
- n = 1, l = 0, Lyman series (ultraviolet)
- n = 2, l = ħ, Balmer series (visible) Wavelength vary from 400 to 700 nm
- n = 3, l = 2ħ, Ritz-Paschen series (short wave infrared)
- n = 4, l = 3ħ, Pfund series (long wave infrared)
Addition of quantized angular momenta
- For more details on this topic, see Angular momentum coupling.
which is the sum of two individual quantized angular momenta and ,
the quantum number
associated with its magnitude can range from to in integer steps
where and are quantum numbers corresponding to the magnitudes of the individual angular momenta.
Total angular momentum of an electron in the atom
Due to the spin-orbit interaction in the atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin. These therefore change over time. However the total angular momentum J does commute with the Hamiltonian and so is constant. J is defined through
stand for , and .
The quantum numbers describing the system, which are constant over time, are now j and
, defined through the action of J on the wavefunction
So that j is related to the norm of the total angular momentum and
to its projection along a specified axis.
As with any angular momentum in quantum mechanics, the projection of J along other axes cannot be co-defined with
, because they do not commute.
Relation between new and old quantum numbers
- For more details on this topic, see Quantum number#Quantum numbers with spin-orbit interaction.
Furthermore, the eigenvectors of j, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of l, ml and ms.
List of angular momentum quantum numbers
- Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number
- orbital angular momentum quantum number
- magnetic quantum number, related to the orbital momentum quantum number
- total angular momentum quantum number
History
The azimuthal quantum number was carried over from the Bohr model of the atom. The Bohr model was derived from spectroscopic analysis of the atom in combination with the Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. To simplify the mathematics, orbits were considered as oscillating charges in one dimension and so described as "pendulum" orbits. In three-dimensions the orbit becomes spherical without any nodes crossing the nucleus, similar to a jump rope that oscillates in one large circle.See also
- Quantum number
- Principal quantum number
- Magnetic quantum number
- Spin quantum number
- Total angular momentum quantum number
- Basic quantum mechanics
External links
- Development of the Bohr atom
- NOTE ON "PENDULUM" ORBITS IN ATOMIC MODELS
- Pictures of atomic orbitals
- Detailed explanation of the Orbital Quantum Number l
- The azimuthal equation explained
Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc.
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An atomic orbital is a mathematical description of the region in which an electron may be found around a single atom.[1] Specifically, atomic orbitals are the possible quantum states of the individual electrons in the electron cloud around a single atom.
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angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
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In atomic physics, the principal quantum number symbolized as n is the first of a set of quantum numbers (which includes: the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) of an atomic orbital.
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Spectroscopic notation provides various ways to specify atomic ionization states, as well as atomic and molecular orbitals.
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Ionization states
Spectroscopists customarily refer to a given ionization state of a given element by giving the element's symbol followed by a Roman..... Click the link for more information.
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers (the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) which describe the unique quantum state of an electron and is designated by
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In atomic physics, the spin quantum number is a quantum number that parametrizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle.
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A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e.
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An atomic orbital is a mathematical description of the region in which an electron may be found around a single atom.[1] Specifically, atomic orbitals are the possible quantum states of the individual electrons in the electron cloud around a single atom.
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Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to
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angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
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Planck constant (denoted ) is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory.
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operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra,
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“Quanta” redirects here. For other uses, see Quantum (disambiguation).
Development of quantum theory
Quantum theory, the branch of physics which is based on quantization, began in 1900 when Max Planck published his theory explaining the..... Click the link for more information.
An atomic orbital is a mathematical description of the region in which an electron may be found around a single atom.[1] Specifically, atomic orbitals are the possible quantum states of the individual electrons in the electron cloud around a single atom.
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A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e.
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associated Legendre functions are the canonical solutions of the general Legendre equation
or
where the indices and m
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or
where the indices and m
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Quantization is the procedure of constraining something from a continuous set of values (such as the real numbers) to a discrete set (such as the integers). Quantization in specific domains is discussed in:
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In atomic physics, the principal quantum number symbolized as n is the first of a set of quantum numbers (which includes: the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) of an atomic orbital.
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In physics, the Lyman series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number referring to the energy level of the electron).
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Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom.
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In physics, the Paschen series (also called Ritz-Paschen series) is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n ≥ 4 to n = 3, where n
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In physics, the Pfund series is a series of absorption or emission lines of atomic hydrogen. The lines were experimentally discovered in 1924 by August Herman Pfund, and correspond to the electron jumping between the fifth and higher energy levels of the hydrogen atom.
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angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case it is useful to couple the spin and orbit angular momentum of the particle.
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Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc.
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