Physics:Radial Excitation Number
In quantum mechanics and hadron spectroscopy, the radial excitation number (typically denoted $n$, $n_r$, or $N$) [1] is a quantum number that quantifies the number of radial nodes in the wave function of a bound state, or the level of excitation along the radial coordinate of a potential.
In atomic physics, it determines the nodal structure of the radial wave function and relates directly to the principal quantum number. In high-energy physics and meson spectroscopy, it classifies radially excited states (such as radial excitations of quarkonium and light mesons) and parameterizes daughter trajectories in Regge theory.
Atomic and quantum mechanical definition
In a spherically symmetric central potential $V(r)$, the three-dimensional time-independent Schrödinger equation separates into angular and radial components:
$$\psi(r, \theta, \phi) = R(r) Y_{\ell}^{m}(\theta, \phi)$$
where $Y_{\ell}^{m}(\theta, \phi)$ are the spherical harmonics parameterized by orbital angular momentum quantum number $\ell$ and magnetic quantum number $m$.
The radial wave function $R(r)$ satisfies the one-dimensional radial equation:
$$-\frac{\hbar^2}{2\mu} \frac{d^2 u(r)}{dr^2} + \left[ V(r) + \frac{\hbar^2 \ell(\ell + 1)}{2\mu r^2} \right] u(r) = E u(r)$$
where $u(r) = r R(r)$ and $\mu$ is the reduced mass.
The radial quantum number $n_r \in \{0, 1, 2, \dots\}$ counts the number of nodes (zeros at finite, non-zero values of $r$) in $u(r)$. In the hydrogen atom and hydrogen-like systems with a pure Coulomb potential, the principal quantum number $n$ is defined by:
$$n = n_r + \ell + 1$$
In this convention, the ground state for a given $\ell$ has $n_r = 0$.
Hadron spectroscopy
In hadron physics and quark model phenomenology, bound states of quarks (such as mesons $q\bar{q}$ and baryons $qqq$) exhibit spectra corresponding to both orbital excitations (increasing angular momentum $L$) and radial excitations (increasing $n$ or $n_r$).
Meson radial excitations
Mesons sharing identical quantum numbers $J^{PC}$ (total angular momentum $J$, parity $P$, and charge conjugation $C$) but differing in mass frequently correspond to successive radial excitations of the ground state.
For example, in the vector meson sector ($I^G J^{PC} = 1^+ 1^{--}$):
$\rho(770)$ represents the ground state ($1\,^3S_1$, $n = 1$).
$\rho(1450)$ is predominantly interpreted as the first radial excitation ($2\,^3S_1$, $n = 2$).
$\rho(1700)$ incorporates mixed orbital-radial contributions ($D$-wave and higher radial excitations).
Similarly, in charmonium ($c\bar{c}$):
$J/\psi(1S)$ is the $1\,^3S_1$ ground state.
$\psi(2S)$ or $\psi(3686)$ is the first radial excitation ($2\,^3S_1$).
Regge trajectories
In relativistic string models, flux tube models, and Regge theory, squared hadron masses $M^2$ follow approximately linear trajectories when plotted against angular momentum $L$ and radial excitation number $n$:
$$M^2 \approx a \cdot n + b \cdot L + c$$
where:
$a$ represents the radial Regge trajectory slope,
$b = 2\pi\sigma$ denotes the orbital Regge trajectory slope (with string tension $\sigma$),
$c$ is a constant intercept corresponding to the ground state.
While semiclassical string quantization and certain holographic AdS/QCD models predict degenerate slopes ($a \approx b$), empirical hadron data analyzed across meson families indicate distinct radial trajectories, often with $a \neq b$.
See also
References
- ↑ Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Pearson Prentice Hall. p. 141. ISBN 978-0131118928.
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