Principal quantum number: Difference between revisions

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The '''principal quantum number''', usually designated by ''n'', is a [[quantum number]] appearing in the description of the [[electron|electronic structure]] of [[atom]]s. The number arises naturally in the solution of the [[Schrödinger equation]] for [[hydrogen-like atom]]s. It is a positive integral number, ''n'' = 1, 2, 3, …,  labeling [[atomic shell]]s.
The '''principal quantum number''', usually designated by ''n'', appears in the description of the [[electron|electronic structure]] of [[atom]]s. The [[quantum number]] arises naturally in the solution of the [[Schrödinger equation]] for [[hydrogen-like atom]]s. It is a positive integral number,   ''n'' = 1, 2, 3, …, that indexes [[atomic shell]]s. Historically, atomic shells were indicated by the capital letters K, L, M, for ''n=1,2,3'', respectively, but this usage is dying out.


In the [[Bohr]]-[[Sommerfeld]] ("old") quantum theory, the electron in a hydrogen-like (one-electron) atom moves in an elliptic orbit. The principal quantum number appears in this theory at two places: in the energy ''E'' of the electron and in the length ''a'' of the major axis of the ellipse.
In the [[Bohr]]-[[Sommerfeld]] ("old") quantum theory, the electron in a hydrogen-like (one-electron) atom moves in elliptic orbits. The principal quantum number appears in this theory at two places: in the energy ''E''<sub>''n''</sub> of the electron and in the length ''a''<sub>''n''</sub> of the major semiaxis of the ellipse described by the ''n''th orbit,
:<math>
:<math>
E = -\frac{m_e}{2} \left( \frac{Z\,e^2}{4\pi\epsilon_0 \hbar \;n}\right)^2 \quad\hbox{and}\quad
E_n = -\frac{m_e}{2} \left( \frac{Z\,e^2}{4\pi\epsilon_0 \hbar \;n}\right)^2  
a =  \frac{4\pi \epsilon_0\; n^2 \hbar^2}{m_e \,Z\, e^2}
\equiv -\frac{Z^2}{n^2}\,\mathrm{Ry}
\quad\hbox{and}\quad
a_n =  \frac{4\pi \epsilon_0\; n^2 \hbar^2}{m_e \,Z\, e^2},
</math>
</math>
Here ''m''<sub>''e''</sub> is the mass of the electron, ''e'' is the charge of the electron, ''Ze'' is the charge of the nucleus,  &epsilon;<sub>0</sub> is the [[electric constant]], and <math>\hbar</math> is [[Planck constant|Planck's reduced constant]]. In the "new" quantum mechanics the energy ''E'' of a hydrogen-like atom is exactly the same, but an electron orbit is replaced by an [[atomic orbital|electron orbital]] that has no radius. However, in the new quantum theory ''a'' appears as the expectation value of ''r'' (the length of the position vector of the electron).
where Ry is the [[Rydberg constant|Rydberg energy]] for infinite nuclear mass (= 13.605&thinsp;6923 eV). Further, ''m''<sub>''e''</sub> is the mass of the electron, &minus;''e'' is the charge of the electron, ''Ze'' is the charge of the nucleus,  &epsilon;<sub>0</sub> is the [[electric constant]], and <math>\hbar</math> is [[Planck constant|Planck's reduced constant]].  


Strictly speaking, the principal quantum number is not defined for many-electron atoms. However, in a fairly good approximate description (central field and independent particles) of the many-electron atom, the principal quantum number does appear and hence ''n'' is a label that is often applied to many-electron atoms as well.  
In the "new" [[quantum mechanics]] (of [[Heisenberg]], [[Schrödinger]], and others) the energy ''E''<sub>''n''</sub> of a bound electron in a hydrogen-like atom is exactly the same, but an electron orbit is replaced by an [[atomic orbital|electron orbital]] that has no radius. However, in the new quantum theory the same expression for ''a''<sub>''n''</sub> appears as the [[expectation value]] of ''r'' (the length of the position vector of the electron) with respect to a state with principal quantum number ''n''. That is, quantum mechanics gives the same measure for the "size" of a one-electron atom  (in state ''n'') as the old quantum theory.
 
Strictly speaking, the principal quantum number is not defined for many-electron atoms. However, in a fairly good approximate description (central field plus independent-particle model) of the many-electron atom, the principal quantum number does appear and hence ''n'' is a label that is often applied to many-electron atoms as well.


==Azimuthal and magnetic quantum numbers==
==Azimuthal and magnetic quantum numbers==

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The principal quantum number, usually designated by n, appears in the description of the electronic structure of atoms. The quantum number arises naturally in the solution of the Schrödinger equation for hydrogen-like atoms. It is a positive integral number, n = 1, 2, 3, …, that indexes atomic shells. Historically, atomic shells were indicated by the capital letters K, L, M, for n=1,2,3, respectively, but this usage is dying out.

In the Bohr-Sommerfeld ("old") quantum theory, the electron in a hydrogen-like (one-electron) atom moves in elliptic orbits. The principal quantum number appears in this theory at two places: in the energy En of the electron and in the length an of the major semiaxis of the ellipse described by the nth orbit,

where Ry is the Rydberg energy for infinite nuclear mass (= 13.605 6923 eV). Further, me is the mass of the electron, −e is the charge of the electron, Ze is the charge of the nucleus, ε0 is the electric constant, and is Planck's reduced constant.

In the "new" quantum mechanics (of Heisenberg, Schrödinger, and others) the energy En of a bound electron in a hydrogen-like atom is exactly the same, but an electron orbit is replaced by an electron orbital that has no radius. However, in the new quantum theory the same expression for an appears as the expectation value of r (the length of the position vector of the electron) with respect to a state with principal quantum number n. That is, quantum mechanics gives the same measure for the "size" of a one-electron atom (in state n) as the old quantum theory.

Strictly speaking, the principal quantum number is not defined for many-electron atoms. However, in a fairly good approximate description (central field plus independent-particle model) of the many-electron atom, the principal quantum number does appear and hence n is a label that is often applied to many-electron atoms as well.

Azimuthal and magnetic quantum numbers

An atomic shell consists of atomic subshells that are labeled by the azimuthal quantum number, commonly denoted by l. For a given atomic shell of principal quantum number n, l runs from 0 to n−1, as follows from the solution of the Schrödinger equation. In total, an atomic shell with quantum number n consists of n subshells and

atomic orbitals. An atomic subshell consists of 2l+1 atomic orbitals labeled by the magnetic quantum number, almost invariably denoted by m. For given l, the magnetic quantum number runs over 2l+1 values: m = −l, −l+1, …, l−1, l.

For historical reasons the orbitals of certain azimuthal quantum number l are denoted by letters: s, p, d, f, g, for l = 0, 1, 2, 3, 4, respectively. For instance an atomic orbital with n = 4 and l = 3 is written as 4d. If the n = 4, l = 2 subshell is occupied k times (there are k electrons in the 4d orbital), we indicate this by writing (4d)k. Any atomic orbital can be occupied at most twice, once with electron spin up and once with spin down. Hence a subshell can accommodate at most 2(2l+1) electrons. If it does, the subshell is called closed. If the atomic shell n contains 2n2 electrons, it is also called closed. For instance the noble gas neon in its lowest energy state has the electron configuration (1s)2(2s)2(2p)6, that is, all its shells and subshells are closed.