Azimuthal quantum number

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The Azimuthal quantum number (or orbital/total 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.

1 Derivation
2 Addition of quantized angular momenta
3 List of angular momentum quantum numbers
4 History
5 See also
6 External links

[edit] Derivation

There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, l, m_l, 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 $\mathbf{}l$ is described by the following equation:

$\mathbf{L} = \hbar\sqrt{l(l+1)}$

where h is Planck's constant and h-bar is Planck's reduced constant, also called Dirac's constant. 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:

$l$	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		fundamental
4	g	18		(letter following F)
5	h	22
6	i	26

A mnemonic for the order of the "shells" is some poor damn fool. 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 m_l (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 nlm_l orbital can be occupied by two electrons with opposing spins (given by the quantum number m_s), 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 ellipticity of the probability cloud and 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 trough which has zero magnitude. A sine wave has a portion with a positive magnitude, a portion with a negative magnitude, and a node, 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:

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)

[edit] Addition of quantized angular momenta

Given a quantized total angular momentum $\overrightarrow{j}$ which is the sum of two individual quantized angular momenta $\overrightarrow{l_1}$ and $\overrightarrow{l_2}$ ,

$\overrightarrow{j} = \overrightarrow{l_1} + \overrightarrow{l_2}$

the quantum number $j$ associated with its magnitude can range from $| l 1 - l 2 |$ to $l 1 + l 2$ in integer steps where $l 1$ and $l 2$ are quantum numbers corresponding to the magnitudes of the individual angular momenta.

[edit] 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

[edit] History

The azimuthal quantum number was carried over from the Bohr model of the atom. The Bohr model was derived as was all of quantum theory from spectroscopic analysis of the atom in combination with the Rutherford atom. This is due to the fact that each atom of each element produces a unique quantum pattern of spectral lines when light from each different kind of element is passed through a prism. Whenever an electron drops to a lower energy level or lower n-orbital, it emits a photon which shows up on the spectrum according to its wavelength. The first orbital n of the Bohr atom, called the K-shell at times, is in the ultraviolet spectrum and was therefore discovered after the Balmer series which is in the visible light spectrum. Bohr argued that the angular momentum in any orbit n was nKh, where h is Planck's constant and K is some multiplying factor, the same for all the orbits, which was later determined to be 1/2 $π$ . The lowest quantum level therefore had an angular momentum of zero. Because orbits were one-dimensionalized in mathematics for simplicity to oscillating charges, this was originally described as a "pendulum" orbit because it was linear. However in three-dimensions the orbit becomes spherical without any nodes crossing the nucleus as in the case of a vibrating string on a musical instrument that were to oscillate in one large circle (similar to a jump rope) thereby creating a standing wave that did not intersect with the nucleus.