27 May 2025
Atomic Spectra
Atomic models, hydrogen wavefunctions, angular momentum, coupling schemes, field splittings, periodic structure, and characteristic X-rays.
Atomic spectra are discrete because bound atomic states have discrete energies. The route from Rutherford scattering to the hydrogen Schrödinger equation also shows why an orbit is not a quantum state.
Rutherford scattering and the nuclear atom
For a projectile of charge $ze$ scattered by a nucleus of charge $Ze$, the repulsive Coulomb potential is
\[V(r)=\frac{Zze^2}{4\pi\epsilon_0r}\equiv\frac{K}{r}.\]Conservation of angular momentum gives $L=\mu vb$, where $b$ is the impact parameter and $\mu$ is the projectile-nucleus reduced mass. With $u=1/r$, Binet’s equation for the outward force $K/r^2$ is
\[\frac{d^2u}{d\phi^2}+u=-\frac{\mu K}{L^2}.\]Its hyperbolic solution can be written
\[r=\frac{p}{e\cos\phi-1},\qquad p=\frac{L^2}{\mu K},\qquad e^2=1+\frac{2EL^2}{\mu K^2}.\]The asymptotes satisfy $\cos\phi_\infty=1/e$. Their geometry gives $\theta=\pi-2\phi_\infty$, so
\[\cot\frac{\theta}{2}=\tan\phi_\infty =\sqrt{e^2-1} =\frac{2Eb}{K},\]where $E=\mu v^2/2$ and $L=\mu vb$ were used in the last step. Therefore
\[b=\frac{K}{2E}\cot\frac{\theta}{2},\qquad E=\frac12\mu v^2.\]Particles incident in the annulus $2\pi b\,db$ enter the solid angle $d\Omega=2\pi\sin\theta\,d\theta$. Hence
\[\frac{d\sigma}{d\Omega} =\frac{b}{\sin\theta}\left\lvert\frac{db}{d\theta}\right\rvert =\left(\frac{K}{4E}\right)^2\csc^4\frac{\theta}{2}.\]The observed large-angle events require charge and most of the mass to occupy a region far smaller than the atom; this is the nuclear model inferred from Rutherford’s experiment.
Bohr hydrogen atom and reduced mass
For a circular orbit, Coulomb attraction supplies centripetal force:
\[\frac{\mu v^2}{r}=\frac{e^2}{4\pi\epsilon_0r^2},\]where nuclear motion is included through the reduced mass
\[\mu=\frac{m_eM}{m_e+M}.\]Bohr quantization, or equivalently the standing-wave condition $2\pi r=n\lambda$ with $\lambda=h/(\mu v)$, gives
\[\mu vr=n\hbar.\]Eliminating $v$ produces
\[r_n=\frac{4\pi\epsilon_0\hbar^2}{\mu e^2}n^2,\]and the total energy is
\[E_n=\frac12\mu v^2-\frac{e^2}{4\pi\epsilon_0r_n} =-\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2}\frac1{n^2}.\]The same quantization condition is the de Broglie standing-wave relation because $\lambda_{dB}=h/(\mu v)$ and $2\pi r=n\lambda_{dB}$. For emission $n_i\to n_f<n_i$,
\[\frac1\lambda=R_M\left(\frac1{n_f^2}-\frac1{n_i^2}\right),\qquad R_M=R_\infty\frac{\mu}{m_e}.\]Thus isotope spectra differ slightly because $\mu$ depends on nuclear mass. The Ritz combination principle follows immediately: if $\tilde\nu_{ab}=T_a-T_b$, then $\tilde\nu_{ac}=\tilde\nu_{ab}+\tilde\nu_{bc}$. Bohr’s model explains hydrogenic energies, radii, and the Rydberg law, but not line intensities, fine structure, multielectron atoms, or why an accelerating charge in a stationary orbit does not radiate.
Hydrogen Schrödinger equation
With $V(r)=-e^2/(4\pi\epsilon_0r)$, the time-independent equation is
\[\left[-\frac{\hbar^2}{2\mu}\nabla^2-\frac{e^2}{4\pi\epsilon_0r}\right]\psi=E\psi.\]In spherical coordinates,
\[\nabla^2=\frac1{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) +\frac1{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) +\frac1{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}.\]Put $\psi=R(r)\Theta(\theta)\Phi(\phi)$. Separation of $\phi$ gives
\[\frac{d^2\Phi}{d\phi^2}+m_l^2\Phi=0,\qquad \Phi(\phi+2\pi)=\Phi(\phi),\]so $m_l$ is an integer. The polar equation is
\[\frac1{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right) +\left[l(l+1)-\frac{m_l^2}{\sin^2\theta}\right]\Theta=0,\]whose regular solutions require $l=0,1,2,\ldots$ and $m_l=-l,\ldots,l$. The angular functions combine into $Y_l^{m_l}$. The radial equation is
\[\frac1{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) +\left[\frac{2\mu}{\hbar^2}\left(E+\frac{e^2}{4\pi\epsilon_0r}\right)-\frac{l(l+1)}{r^2}\right]R=0.\]Writing $u=rR$ removes the first derivative:
\[-\frac{\hbar^2}{2\mu}\frac{d^2u}{dr^2} +\left[-\frac{e^2}{4\pi\epsilon_0r}+\frac{\hbar^2l(l+1)}{2\mu r^2}\right]u=Eu.\]For a bound state define
\[\kappa=\frac{\sqrt{-2\mu E}}{\hbar},\qquad \rho=2\kappa r,\]and set $u(\rho)=\rho^{l+1}e^{-\rho/2}v(\rho)$. The factor $\rho^{l+1}$ enforces $u(0)=0$, while $e^{-\rho/2}$ enforces decay as $r\to\infty$. Substitution gives
\[\rho v''+[2(l+1)-\rho]v' +\left[\frac{\mu e^2}{4\pi\epsilon_0\hbar^2\kappa}-(l+1)\right]v=0.\]The series for $v$ must terminate; otherwise its $e^\rho$ growth defeats the decaying prefactor. Therefore
\[\frac{\mu e^2}{4\pi\epsilon_0\hbar^2\kappa}=n_r+l+1\equiv n,\]giving $n=1,2,\ldots$, $l=0,\ldots,n-1$, and the same $E_n$ as the Bohr result. The normalized ground state is
\[\psi_{100}(r)=\frac{e^{-r/a_\mu}}{\sqrt{\pi a_\mu^3}},\qquad a_\mu=\frac{4\pi\epsilon_0\hbar^2}{\mu e^2}.\]The electric-dipole matrix element $\langle n^{\prime}l^{\prime}m_l^{\prime}\rvert\mathbf r\lvert nlm_l\rangle$ and the angular properties of $\mathbf r$ give the selection rules
\[\boxed{\Delta l=\pm1,\qquad \Delta m_l=0,\pm1},\]with parity changing between the two states.
Orbital and spin angular momentum
The vector model represents
\[L^2=l(l+1)\hbar^2,\quad L_z=m_l\hbar, \qquad S^2=s(s+1)\hbar^2,\quad S_z=m_s\hbar.\]For an electron $s=1/2$ and $m_s=\pm1/2$. Its magnetic moments are
\[\boldsymbol\mu_L=-\frac{e}{2m_e}\mathbf L=-\frac{\mu_B}{\hbar}\mathbf L,\] \[\boldsymbol\mu_S=-g_s\frac{\mu_B}{\hbar}\mathbf S,qquad \mu_B=\frac{e\hbar}{2m_e},\quad g_s\simeq2.\]The constant $\mu_B=e\hbar/(2m_e)$ is the Bohr magneton. The Stern-Gerlach force $F_z=\mu_z\,dB_z/dz$ separates an atomic beam into discrete components, demonstrating space quantization. In a uniform field, $d\mathbf L/dt=\boldsymbol\mu_L\times\mathbf B$ gives Larmor precession with
\[\omega_L=\frac{eB}{2m_e}.\]Adding angular momenta gives $\mathbf J=\mathbf L+\mathbf S$ and
\[j=\lvert l-s\rvert,\lvert l-s\rvert+1,\ldots,l+s.\]Since $2\mathbf L\cdot\mathbf S=J^2-L^2-S^2$, a spin-orbit Hamiltonian $H_{so}=\xi(r)\mathbf L\cdot\mathbf S$ shifts a level by
\[\Delta E_{so}=\frac{\xi}{2}\hbar^2[j(j+1)-l(l+1)-s(s+1)].\]For weak residual interactions in light atoms, individual orbital momenta form $\mathbf L=\sum_i\mathbf l_i$, spins form $\mathbf S=\sum_i\mathbf s_i$, and then $\mathbf J=\mathbf L+\mathbf S$: this is LS coupling. In heavy atoms, each $\mathbf j_i=\mathbf l_i+\mathbf s_i$ forms first and $\mathbf J=\sum_i\mathbf j_i$: this is jj coupling.
Exchange symmetry, Pauli principle, and periodic structure
For identical particles, exchange cannot produce a physically distinguishable state:
\[\Psi(1,2)=\pm\Psi(2,1).\]Integer-spin particles use the symmetric sign; electrons are fermions and require the antisymmetric sign. If two electrons occupied the same one-particle state $a$,
\[\Psi=\frac1{\sqrt2}[\psi_a(1)\psi_a(2)-\psi_a(2)\psi_a(1)]=0.\]This is Pauli exclusion: no two electrons in an atom share all four quantum numbers. Shell capacity follows by counting $m_l$ and spin states:
\[N_n=2\sum_{l=0}^{n-1}(2l+1)=2n^2.\]Successive filling of subshells, modified by their energies and exchange, produces the periodic table.
Zeeman, Paschen-Back, and Stark effects
In a weak magnetic field, $J$ remains coupled and
\[\Delta E_Z=\mu_Bg_Jm_JB,\]where
\[g_J=1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}.\]In a strong field the spin-orbit coupling is overcome. The Paschen-Back shift approaches
\[\Delta E_{PB}=\mu_BB(m_l+g_sm_s).\]An electric field $\mathbf E=E\hat z$ adds $H^{\prime}=eEz$ for an electron in the nuclear potential. The first-order shift is $\Delta E^{(1)}=eE\langle z\rangle$. It vanishes for a nondegenerate parity eigenstate, but hydrogen states with the same $n$ are degenerate and mix, producing a linear Stark effect. Nondegenerate states begin with the quadratic shift
\[\Delta E^{(2)}=-\frac12\alpha E^2,\]where $\alpha$ is the static polarizability.
Characteristic X-ray spectra
An incident electron can remove an inner-shell electron. When an outer electron fills the vacancy, a photon carries the energy difference:
\[h\nu=E_{n_i}-E_{n_f}.\]Screening replaces $Z$ by $Z-\sigma$, so the hydrogenic approximation gives Moseley’s form
\[\sqrt\nu=C(Z-\sigma).\]The $K_\alpha$ line is predominantly a $2p\to1s$ transition and the $K_\beta$ line a $3p\to1s$ transition, consistent with $\Delta l=\pm1$. These discrete characteristic lines lie on the continuous bremsstrahlung spectrum.
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