6.3 Quantum Numbers
The Quantum Numbers
Modern quantum theory tells us that the various allowed states of existence of the electron in the hydrogen atom correspond to different standing wave patterns. In the preceding lesson we showed examples of standing waves that occur on a vibrating guitar string. The wave patterns of electrons in an atom are different in two important ways:
- Instead of indicating displacement of a point on a vibrating string, the electron waves represent the probability that an electron will manifest itself (appear to be located) at any particular point in space. (Note carefully that this is not the same as saying that “the electron is smeared out in space”; at any given instant in time, it is either at a given point or it is not.)
- The electron waves occupy all three dimensions of space, whereas guitar strings vibrate in only two dimensions.
These 
videos made for the eChem1a series at UC-Berkeley provide an excellent overview of atomic orbitals.
This one (8 min) is especially helpful in showing how the three quantum numbers n, l, and m arise from the wave function of the atom.
Other videos
Orbital energies and the principal quantum number (4 min)
Quantum Numbers, Atomic Orbitals, and Electron Configurations
s Orbitals of the hydrogen atom
(3½ min)
Multi-electron atoms and shielding
(4½ min)
What is electron shielding in atoms
Radial nodes of orbitals (quiz)
How to Determine Number of Angular Nodes, Radial Nodes, and Total Nodes of Orbitals Examples
Restrictions on quantum numbers
How To Determine The 4 Quantum Numbers From an Element or a Valence Electron
Electron spin, magnetism (7 min)
Aside from this, the similarities are striking. Each wave pattern is identified by an integer number n, which in the case of the atom is known as the principal quantum number. The value of n tells how many peaks of amplitude (antinodes) exist in that particular standing wave pattern; the more peaks there are, the higher the energy of the state.
Figure 6.32 Electron Density vs. Distance from Nucleus
The three simplest orbitals of the hydrogen atom are depicted above in pseudo-3D, in cross-section, and as plots of probability (of finding the electron) as a function of distance from the nucleus. The average radius of the electron probability is shown by the blue circles or plots in the two columns on the right. These radii correspond exactly to those predicted by the Bohr model.
This formula was actually part of Bohr’s original theory, and is still applicable to the hydrogen atom, although not to atoms containing two or more electrons. In the Bohr model, each value of n corresponded to an orbit of a different radius. The larger the orbital radius, the higher the potential energy of the electron; the inverse square relationship between electrostatic potential energy and distance is reflected in the inverse square relation between the energy and n in the above formula.
Although the concept of a definite trajectory or “orbit” of the electron is no longer tenable, the same orbital radii that relate to the different values of n in Bohr’s theory now have a new significance: they give the average distance of the electron from the nucleus. As you can see from the figure, the averaging process must encompass several probability peaks in the case of higher values of n. The spatial distribution of these probability maxima defines the particular orbital.
This physical interpretation of the principal quantum number as an index of the average distance of the electron from the nucleus turns out to be extremely useful from a chemical standpoint, because it relates directly to the tendency of an atom to lose or gain electrons in chemical reactions.
The angular momentum quantum number
The electron wave functions that are derived from Schrödinger’s theory are characterized by several quantum numbers. The first one, n, describes the nodal behavior of the probability distribution of the electron, and correlates with its potential energy and average distance from the nucleus as we have just described.
The angular momentum quantum number is conventionally represented by lower-case L: l. In order to avoid confusion with the numeral 1, it is ordinarily italized:
l. Additional clarity, which we employ here, is provided by using the scripted form: ℓ.
Figure 6.33 Probability Density vs. shapes of Orbitals
The theory also predicts that orbitals having the same value of n can differ in shape and in their orientation in space. The quantum number ℓ, known as the angular momentum quantum number, determines the shape of the orbital. (More precisely, ℓ determines the number of angular nodes, that is, the number of regions of zero probability encountered in a 360° rotation around the center.)
When ℓ = 0, the orbital is spherical in shape.
If ℓ = 1, the orbital is elongated into something resembling a figure-8 shape, and higher values of ℓ correspond to still more complicated shapes— but note that the number of peaks in the radial probability distributions (below) decreases with increasing ℓ. The possible values that ℓ can take are strictly limited by the value of the principal quantum number; ℓ can be no greater than n – 1. This means that for
n = 1, ℓ can only have the single value zero which corresponds to a spherical orbital. For historical reasons, the orbitals corresponding to different values of ℓ are designated by letters, starting with s for ℓ = 0, p for ℓ = 1, d for ℓ = 2, and f for ℓ = 3.
in which z is the nuclear charge of the atom, which of course is unity for hydrogen.
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The magnetic quantum number
An s-orbital, corresponding to ℓ = 0, is spherical in shape and therefore has no special directional properties. The probability cloud of a p orbital is aligned principally along an axis extending along any of the three directions of space. The additional quantum number m is required to specify the particular direction along which the orbital is aligned.
“Direction in space” has no meaning in the absence of a force field that serves to establish a reference direction. For an isolated atom there is no such external field, and for this reason there is no distinction between the orbitals having different values of m. If the atom is placed in an external magnetic or electrostatic field, a coordinate system is established, and the orbitals having different values of m will split into slightly different energy levels. This effect was first seen in the case of a magnetic field, and this is the origin of the term magnetic quantum number.
In chemistry, however, electrostatic fields are much more important for defining directions at the atomic level because it is through such fields that nearby atoms in a molecule interact with each other. The electrostatic field created when other atoms or ions come close to an atom can cause the energies of orbitals having different direction properties to split up into different energy levels; this is the origin of the colors seen in many inorganic salts of transition elements, such as the blue color of copper sulfate.
The quantum number m can assume 2ℓ + 1 values for each value of ℓ, from –ℓ through 0 to +ℓ. When ℓ = 0 the only possible value of m will also be zero, and for the p orbital (ℓ = 1), m can be –1, 0, and +1. Higher values of ℓ introduce more complicated orbital shapes which give rise to more possible orientations in space, and thus to more values of m.
Electron spin and the Exclusion Principle
Certain fundamental particles have associated with them a magnetic moment that can align itself in either of two directions with respect to an external magnetic field. The electron is one such particle, and the direction of its magnetic moment is called its spin.
The mechanical analogy implied by the term spin is easy to visualize, but should not be taken literally. Physical rotation of an electron is meaningless. However, the coordinates of the electron’s wave function can be rotated mathematically; when this is done, it is found that a rotation of 720° is required to restore the function to its initial value— rather weird, considering that a 360° rotation will leave any extended body unchanged! Electron spin is basically a relativistic effect in which the electron’s momentum distorts local space and time. It has no classical counterpart and thus cannot be visualized other than through mathematics.
Exclusion principle and Hund’s rule
UC-Berkeley, 6 min)
How the Schrödinger equation predicts the exclusion principle
(TinaHuang, 4 min)
Schroedinger Equation & Pauli Exclusion principle
A basic principle of modern physics states that for particles such as electrons that possess half-integral values of spin, no two of them can be in identical quantum states within the same system. The quantum state of a particle is defined by the values of its quantum numbers, so what this means is that no two electrons in the same atom can have the same set of quantum numbers. This is known as the Pauli exclusion principle, named after the German physicist Wolfgang Pauli (1900-1958, Nobel Prize 1945).
The best non-mathematical explanation of the exclusion principle that I have come across is Phil Fraundorf’s Candle Dances and Atoms page at U. Missouri-St. Louis.
The exclusion principle was discovered empirically and was placed on a firm theoretical foundation by Pauli in 1925. A complete explanation requires some familiarity with quantum mechanics, so all we will say here is that if two electrons possess the same quantum numbers n, l, m and s (defined below), the wave function that describes the state of existence of the two electrons together becomes zero, which means that this is an “impossible” situation.
A given orbital is characterized by a fixed set of the quantum numbers n, ℓ, and m. The electron spin itself constitutes a fourth quantum number s, which can take the two values +1 and –1. Thus a given orbital can contain two electrons having opposite spins, which “cancel out” to produce zero magnetic moment. Two such electrons in a single orbital are often referred to as an electron pair.
More on the Pauli exclusion principle from answers.com
If it were not for the exclusion principle, the atoms of all elements would behave in the same way, and there would be no need for a science of Chemistry!
As we have seen, the lowest-energy standing wave pattern the electron can assume in an atom corresponds to n=1, which describes the state of the single electron in hydrogen, and of the two electrons in helium. Since the quantum numbers m and ℓ are zero for n=1, the pair of electrons in the helium orbital have the values (n, l, m, s) = (1,0,0,+1) and (1,0,0,–1)— that is, they differ only in spin. These two sets of quantum numbers are the only ones that are possible for a n=1 orbital. The additional electrons in atoms beyond helium must go into higher-energy (n>1) orbitals.
Electron wave patterns corresponding to these greater values of n are concentrated farther from the nucleus, with the result that these electrons are less tightly bound to the atom and are more accessible to interaction with the electrons of neighboring atoms, thus influencing their chemical behavior. If it were not for the Pauli principle, all the electrons of every element would be in the lowest-energy n=1 state, and the differences in the chemical behavior the different elements would be minimal. Chemistry would certainly be a simpler subject, but it would not be very interesting!