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Quantum number

Quantum numbers are quantities used in describing the state of a quantum mechanical system. [A full set corresponds to a complete set of commuting observables.]

For example, the state of an electron in a hydrogen atom can be fully described by determining four quantum numbers, traditionally known as the principal ($n$), azimuthal ($l$), magnetic ($m_{l}$) and spin ($m_{s}$) quantum numbers. The first three numbers are related to electron's movement and distribution in space.

The spin quantum number is an internal quantum number describing degrees of freedom with no classical counterpart. Other elementary particles are described by other internal quantum numbers... [?]

According to Pauli exclusion principle, two fermions cannot share the same quantum numbers.

Quantum numbers are "good" when they are conserved in the dynamics of a quantum system. Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian.

General properties

Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian, quantities that can be known with precision at the same time as the system's energy. Specifically, observables that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues $a$ and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together.

Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers. Another, perhaps more quantum, aspect is that a measurement returns eigenvalues according to Born rule

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian, H. There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. Energy is not necessarily the best choice, and not always include. E.g. for a free particle, the momentum might be a better one, and it determines the energy by dispersion.

As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations. This is not necessarily so much about the choice of the basis states, but how they are labeled. More accurately, both the choice and labeling of the basis play a role here.

Other single-electron systems

In other single-electron systems with no spin-orbit coupling the quantum numbers generally differ from those of the hydrogen atom.

• The quantum numbers for a free particle are the three components of momentum. In this case, the quantum numbers are not discrete.
• In the particle-in-a-box model with volume $L^{3}$, the system is not spherically symmetric, and the three quantum numbers $n$,$\ell$,$m_{\ell }$ are replaced by a three-component wavenumber $\mathbf {n} =(n_{x},n_{y},n_{z})$. The components determine the number of nodal surfaces perpendicular to the coordinate axes in each direction.
• In a crystal, the electrons move in the potential field of the atoms, which is periodic and has the same symmetry as the crystal. According to the Bloch theorem, the good quantum numbers in this case are the crystal momentum, which in a infinite crystal is continuous, but restricted to the First Brillouin zone, and the index of the electronic band. There is some freedom in choosing the quantum numbers (See extended vs. reduced zone scheme).

Good quantum number

In quantum mechanics, the eigenvalue $q$ of an observable $O$ is a good quantum number if the observable $O$ is a constant of motion. In other words, the quantum number $q$ is good if the corresponding observable commutes with the Hamiltonian. If the system starts from the eigenstate with an eigenvalue $q$, it remains on that state as the system evolves in time.

Good quantum numbers are often used to label initial and final states in experiments. For example, in particle colliders:

1. Particles are initially prepared in approximate momentum eigenstates; the particle momentum being a good quantum number for non-interacting particles.
2. The particles are made to collide. At this point, the momentum of each particle is undergoing change and thus the particles’ momenta are not a good quantum number for the interacting particles during the collision.
3. A significant time after the collision, particles are measured in momentum eigenstates. Momentum of each particle has stabilized and is again a good quantum number a long time after the collision.

Group theory and quantum numbers

Group of Schrödinger equation formed from the set of hermitean operators which commute with the Hamiltonian. Accidental degeneracies (Laplace-Runge-Lenz), and those grounded in symmetry. The set of degenerate eigenfunctions forms a irrep of the Schrödinger group. If there no accidental degeneracies, then the eigenfunctions of each energy level provide an irreducible representation of the symmetry group. Perturbation can only lift degeneracies if the symmetry group changes.

"In this way group theory provides "good quantum numbers" for any problem in the form of the labels of the representations and the rows within each one."

• Tinkham, Michael (2003-12-17). Group Theory and Quantum Mechanics. Courier Corporation. pp. 31–39. ISBN 978-0-486-43247-2.