|
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron (the others being the principal quantum number, following spectroscopic notation, the magnetic quantum number, and the spin quantum number). It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as ℓ. == Derivation == Connected with the energy states of the atom's electrons are four quantum numbers: ''n'', ''ℓ'', ''m''''ℓ'', and ''m''''s''. These specify the complete, unique quantum state of a single electron in an atom, and make up its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to 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. To aid understanding of this concept of the azimuth, it may also prove helpful to review spherical coordinate systems, and/or other alternative mathematical coordinate systems besides the Cartesian coordinate system. Generally, the spherical coordinate system works best with spherical models, the cylindrical system with cylinders, the cartesian with general volumes, etc. An atomic electron's angular momentum, ''L'', is related to its quantum number ''ℓ'' by the following equation: : where ''ħ'' is the reduced Planck's constant, L2 is the orbital angular momentum operator and is the wavefunction of the electron. The quantum number ''ℓ'' is always a nonnegative integer: 0,1,2,3, etc. (see angular momentum quantization). While many introductory textbooks 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 better to simply use the quantum number ''ℓ''. Atomic orbitals 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: : The letters after the ''f'' sub-shell just follow ''f'' in alphabetical order except ''j'' and those already used. One mnemonic to remember the sequence ''S''. ''P''. ''D''. ''F''. ''G''. ''H''. ... is "Sober Physicists Don't Find Giraffes Hiding In Kitchens Like My Nephew". A few other mnemonics are Smart People Don't Fail, Silly People Drive Fast, silly professors dance funny, Scott picks dead flowers, some poor dumb fool! etc. Each of the different angular momentum states can take 2(2''ℓ'' + 1) electrons. This is because the third quantum number ''m''''ℓ'' (which can be thought of loosely as the quantized projection of the angular momentum vector on the z-axis) runs from −''ℓ'' to ''ℓ'' in integer units, and so there are 2''ℓ'' + 1 possible states. Each distinct ''n'',''ℓ'',''m''''ℓ'' orbital can be occupied by two electrons with opposing spins (given by the quantum number ''ms''), giving 2(2''ℓ'' + 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 ''n''th energy level is 2''n''2. 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 trough, 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 0. In a p orbital, one node traverses the nucleus and therefore ''ℓ'' has the value of 1. ''L'' has the value √2''ħ''. Depending on the value of ''n'', 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'' = √2''ħ'', Balmer series (visible) :''n'' = 3, ''L'' = √6''ħ'', Ritz-Paschen series (near infrared) :''n'' = 4, ''L'' = 2√3''ħ'', Brackett series (short-wavelength infrared) :''n'' = 5, ''L'' = 2√5''ħ'', Pfund series (mid-wavelength infrared). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Azimuthal quantum number」の詳細全文を読む スポンサード リンク
|