Quantum Numbers

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Quantum numbers are what we use to label elementary particles and their values are crucial for the behaviour of particles in experiments.

For example, an electron is defined by the following quantum numbers

• mass: $9.109 \cdot 10^{-31}$ kg
• spin: $\frac{1}{2}$
• electric charge: $-1.602 \cdot 10^{-19}$ C
• weak charge, called weak isospin: $-\frac{1}{2}$ (for the left-handed electron; the right-handed electron has $0$)
• strong charge, called color charge: $0$

These labels determine how a given elementary particle behaves in experiments. The technical term for these labels is quantum numbers.

Concrete

In quantum mechanics a state is labelled by the eigenvalues of a set of commuting observables. Quantum numbers are these eigenvalues, chosen so that the labels are stable in time. An observable $\hat Q$ that commutes with the Hamiltonian is conserved,

$$\frac{d}{dt}\langle \hat Q\rangle = \frac{i}{\hbar}\langle[\hat H, \hat Q]\rangle = 0 \qquad (\hat Q \text{ without explicit time dependence}),$$

and $\hat H$ and $\hat Q$ can be diagonalized together. The eigenvalue of $\hat Q$ in an energy eigenstate then never changes under time evolution. Every such operator comes from a symmetry of the system, and so every conserved quantum number is tied to a symmetry.

Example: the hydrogen atom. In the nonrelativistic hydrogen atom the operators $\hat H$, $\hat{\vec L}^2$, $\hat L_z$ and $\hat S_z$ commute with each other. Their eigenvalues give the quantum numbers

• principal quantum number $n = 1, 2, 3, \ldots$, with energy $E_n = -\frac{13.6\ \mathrm{eV}}{n^2}$,
• orbital angular momentum $l = 0, 1, \ldots, n-1$, with $\hat{\vec L}^2 = \hbar^2 l(l+1)$,
• magnetic quantum number $m_l = -l, \ldots, l$, with $\hat L_z = \hbar m_l$,
• spin projection $m_s = \pm\frac12$, with $\hat S_z = \hbar m_s$.

Counting states gives $\sum_{l=0}^{n-1} 2(2l+1) = 2n^2$ states with energy $E_n$. Once fine structure is included, $\hat L_z$ and $\hat S_z$ no longer commute with $\hat H$, and one uses the total angular momentum $j$ and $m_j$ instead. Which quantum numbers are good depends on which symmetries the Hamiltonian has.

Particle quantum numbers. Elementary particles are labelled by mass, spin, electric charge $Q$, and their charges under the weak and strong interactions. Some quantum numbers are additive: the electric charge of a system is the sum of the charges of its parts. Others are multiplicative, such as parity $P = \pm 1$, where the parity of a system of particles is the product of their intrinsic parities times $(-1)^l$ for relative orbital angular momentum $l$. In the Standard Model the electric charge is related to the third component of weak isospin $T_3$ and the weak hypercharge $Y$ by

$$Q = T_3 + \frac{Y}{2}$$

(in the convention where the hypercharge of the left-handed lepton doublet is $Y = -1$; some books use $Q = T_3 + Y$ instead). For the left-handed electron $T_3 = -\frac12$ and $Y = -1$, for the right-handed electron $T_3 = 0$ and $Y = -2$; both give $Q = -1$ in units of the elementary charge $e$. Conservation laws for quantum numbers decide which reactions are allowed: the decay $n \to p + e^- + \bar\nu_e$ conserves electric charge, baryon number and lepton number.

• http://jakobschwichtenberg.com/what-are-quantum-numbers/
• No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, section "Quantum Numbers"

Abstract

The quantum numbers of a system are labels of representations of its symmetry group $G$ on the Hilbert space. If $G$ acts by a unitary representation that commutes with the Hamiltonian, the Hilbert space decomposes into irreducible representations, and each energy eigenspace is a representation of $G$. A state is then labelled by the irreducible representation it lies in (the eigenvalues of the Casimir operators) and by its position inside that representation (the eigenvalues of a maximal set of commuting generators, a Cartan subalgebra, which are the weights). For $SU(2)$ these are $j$ and $m$. For larger groups such as $SU(3)$ a weight can occur more than once in an irreducible representation, and further labels are needed. See representation theory.

Spacetime quantum numbers. In relativistic quantum theory the symmetry group contains the Poincaré group, or rather its universal cover. Following Wigner, one-particle states form irreducible unitary representations, classified by the Casimir operators. With $\hbar = c = 1$ and signature $(+,-,-,-)$ these are

$$P_\mu P^\mu = m^2, \qquad W_\mu W^\mu = -m^2\, s(s+1) \quad (m > 0),$$

where $W^\mu = \frac12 \epsilon^{\mu\nu\rho\sigma} P_\nu M_{\rho\sigma}$ is the Pauli-Lubanski vector. Mass and spin are therefore representation labels of the Poincaré group. For $m > 0$ the spin $s \in \{0, \frac12, 1, \ldots\}$ labels a representation of the little group $SU(2)$. For $m = 0$ the relevant representations of the little group (the double cover of $ISO(2)$ with trivial action of the translations) are labelled by the helicity $h \in \frac12\mathbb Z$. See Wigner's little groups.

Internal quantum numbers. Charges such as electric charge, colour and weak isospin label representations of internal symmetry groups, which act on the fields at each point and commute with the Poincaré transformations. In the Standard Model the gauge group is $SU(3)\times SU(2)\times U(1)$, and each field transforms in a representation such as $(\mathbf 3, \mathbf 2)_{1/3}$ for the left-handed quark doublet (hypercharge in the convention $Q = T_3 + Y/2$). The Coleman-Mandula theorem states that, under its assumptions (among them finitely many particle types below any given mass and nontrivial scattering), the Lie algebra of symmetries of the S-matrix is a direct sum of the Poincaré algebra and an internal symmetry algebra. This is why spacetime and internal quantum numbers are independent labels. Supersymmetry evades the theorem by allowing a graded Lie algebra, and its generators change spin by $\frac12$.

Discrete quantum numbers such as parity, charge conjugation and time reversal come from discrete symmetries. For a $\mathbb Z_2$ symmetry the eigenvalues are $\pm 1$, which is why these quantum numbers combine multiplicatively, whereas eigenvalues of generators of continuous abelian groups like $U(1)$ add under tensor products.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Lagrangian Formalism
14. Phase Space
15. Hamiltonian Formalism
16. Schrödinger Equation
17. Canonical Quantum Mechanics