7.10: Quantum Numbers

It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.

The energy levels are labeled with an n value, where n = 1, 2, 3, etc. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number.

The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation.

Another quantum number is l, the secondary (angular momentum) quantum number. It is an integer that may take the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. Whereas the principal quantum number, n, defines the general size and energy of the orbital, the secondary quantum number l specifies the shape of the orbital. Orbitals with the same value of l define a subshell.

Orbitals with l = 0 are called s orbitals, and they make up the s subshells. The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f-, g-, and h-orbitals for l = 3, 4, and 5.

The magnetic quantum number, m_l, specifies the relative spatial orientation of a particular orbital. Generally speaking, m_l can be equal to –l, –(l – 1), …, 0, …, (l – 1), l. The total number of possible orbitals with the same value of l (that is, in the same subshell) is 2l + 1. Thus, there is one s-orbital in an s subshell (l = 0), there are three p-orbitals in a p subshell (l = 1), five d-orbitals in a d subshell (l = 2), seven f-orbitals in an f subshell (l = 3), and so forth. The principal quantum number defines the general value of electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies the orientation of the orbital in space.

While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number or s.

The other three quantum numbers, n, l, and m_l are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms.

The fourth quantum number, m_s, is the spin quantum number. Electrons are spinning charges and behave like tiny bar magnets. The two possible spinning motions of the electron are clockwise and counterclockwise. For an electron in an orbital, these two possibilities are indicated by spin quantum numbers, +1/2 for a clockwise spin, and −1/2 for a counterclockwise spin. It is the only quantum number having non-integral values.

This text is adapted from Openstax, Chemistry 2e, Section 6.3: Development of Quantum Theory.

Each electron is described by four quantum numbers, three of which have interrelated values. These three quantum numbers define where in space the electron is most likely to be located.

The principal quantum number, n, denotes the energy level to which an electron belongs. n can have integer values of one, two, three, and so forth. The greater the value of n, the farther the electron is from the nucleus.

The angular momentum, or azimuthal, quantum number, l, describes the shape of the orbital that the electron occupies, which is related to the electron’s angular momentum. 

l can have integer values from 0 to n – 1 for each value of n. When n is one, l is zero. When n is two, l has two possible values — zero and one. When n is three, l can be zero, one, or two. The number of possible l values indicates the number of groups of differently shaped orbitals, or subshells, at the nth energy level.

Subshells are designated by letters associated with the values of l. Zero, one, two, and three correspond to s, p, d, and f orbitals, respectively. These will be discussed in the next lesson. 

The magnetic quantum number, m_l, describes the orientation in space of the orbital that the electron occupies, which is also related to the electron’s angular momentum. m_l can have any integer value between –l and +l, including zero. The number of orientations for a given subshell is equal to 2l + 1.

When n is two, there are two subshells, as l can be zero or one. These are the 2s and 2p subshells, respectively. The 2p subshell has three 2p orbitals, as there are three values for m_l, –1, 0, and +1. 

The fourth quantum number, m_s, is the spin quantum number. Electrons are spinning charges and behave like tiny bar magnets. The two possible spinning motions for the electron are clockwise, indicated by an m_s of +1/2, and counterclockwise, indicated by an m_s of –1/2.