Quantum Numbers

Quantum Numbers

Quantum Numbers describe the conserved quantities in a quantum mechanical system.

More technically, quantum numbers are eigenvalues of operators that commute with the Hamiltonian of a system. The Hamiltonian is an operation which when applied to a system gives the total energy of that system.
So quantum numbers correspond to quantities that can be known exactly at the same time as the system's
energy and their corresponding eigenstates. Together, the set of all quantum numbers of a system characterize the basis state of a system, and in principle can be measured together. [What are eigenstates and eigenvalues? It's part of vector fields and linear algebra]

Most quantum numbers are additive, but some like parity are multiplicative.

For Electrons In Atomic Orbitals

Principal Quantum Number, \(n\) (Shell)
- Can give the energy level of an electron. It also describes the n-th eigenvalue of the Hamiltonian, that is, the energy excluding the term with the angular momentum \(J^2\). So this number only depends on \(r\), the distance from the center, and it increases with it. Hence, electrons are said to be in different shells based on \(n\).
Azimuthal Quantum Number, \(l\) (Subshell)
- Can give the orbital angular momentum (\(L\)) through the relation \(L^2=\hslash^2 l(l+1)\). Values range from \(0\) to \(n-1\).
- The orbitals with \(l = 0,1,2,3\) are named s-orbital, p-orbital, d-orbital and f-orbital respectively.
- It also describes the shape of the orbital.
Magnetic Quantum Number, \(m_l\) (Orbital Orientation)
- Can specify the orbital cloud, or the projection of the orbital angular momentum along an axis (\(L_z\)), conventionally the z-axis. Values range from \(-l\) to \(+l\). It also describes the specific orbital cloud within the orbital.
Magnetic Spin Quantum Number (\(m_s\)) (Electron Spin)
- Can specify the spin angular momentum (\(S\)) of the electron within an orbital. Values range from \(-s\) to \(s\), where \(s\) is the spin of the particle. For electrons, \(s=\frac{1}{2}\), and hence \(m_s=\pm\frac{1}{2}\).
- It also gives the projection of the spin angular momentum along the specified axis (\(S_z = m_s\hslash\)).

For Total Angular Momentum

When one takes the spin-orbital interactions into consideration, the \(L\) and \(S\) operators no longer commute with the Hamiltonian, and their eigenvalues change over time. Hence new quantum numbers should be used. With spin orbital interactions, the total angular momentum \(J\) shifts back and forth between \(L\) and \(S\), while still remaining conserved.

Of a particle

In the presence of spin-orbital interactions, particles cannot be described using the previously mentioned quantum numbers with separate angular momentum and spin angular momentum components. These new quantum numbers should hence be used to represent the same states.

For example, to represent all the quantum states of the electrons for which \(n=2\).

Total Angular Momentum (\(j\)): \(j=|l \pm m_s|\)
Projection of Total Angular Momentum along a specified axis
- \(m_j\) ranges from \(-j\) to \(+j\)
- Like the above, \(m_j=m_l+m_s\) and \(|m_l+m_s| \leq j\)
Parity: It is +1 for states coming from even \(l\) and -1 for states coming from odd \(l\)

Nuclear Angular Momentum

The nucleons have a total angular momentum due to the angular momenta of the individual nucleons, and is denoted by \(I\).
If the total angular momentum of a neutron is \(j_n=l+s\) and that of a proton is \(j_p=l+s\) (where \(s\) for both protons and neutrons happens to be \(\frac{1}{2}\)), the values of the total nuclear angular momentum quantum numbers, \(I\) are \(|j_n-j_p|\), \(|j_n-j_p|+1\), \(|j_n-j_p|+2\), ..., \(|j_n+j_p|-2\), \(|j_n+j_p|-1\), \(|j_n+j_p|\).

Note

The angular orbital momentum of the nucleons are an integer multiples of \(\hslash\), while the intrinsic angular momentum of both neutrons and protons are half integer multiples. It should be clear that the total angular momentum will always give half integer values for nuclei with odd A and integer values for even A. (A being the atomic mass)

In Fundamental Particles

Fundamental particles contain many quantum numbers intrinsic to them. Fundamental particles themselves are quantum states of the standard model of particle physics. So they bear the same relation to the Hamiltonian of the model as the Bohr atom does to it's Hamiltonian. So, quantum numbers represent the symmetries of a model.

It is useful in Quantum Field Theory to distinguish between the symmetries of spacetime and internal symmetries.

Examples of spacetime symmetries include spin (related to rotational symmetry), parity, C-parity and T-parity (related to Poincare Symmetries of spacetime).

Examples of intrinsic symmetries include the baryon number, lepton number and electric charge. For a full list, see Flavour.