## Quantum Mechanics II: Atomic Structure

Three-dimensional problems:  The time-independent Schrödinger equation for three-dimensional problems is given by: $-\frac{\hbar^2}{2m}(\frac{\partial^2\psi(x,y,z)}{\partial x^2}+\frac{\partial^2\psi(x,y,z)}{\partial y^2}+\frac{\partial^2\psi(x,y,z)}{\partial z^2})+U(x,y,z)\psi(x,y,z)=E\psi(x,y,z)$.

Particle in a three-dimensional box:  The wave function for a particle in a cubical box is the product of a function of $x$ only, a function of $y$ only, and a function of $z$ only.  Each stationary state is described by three quantum numbers $(n_X,n_Y,n_Z)$: $E_{n_X,n_Y,n_Z}=\frac{(n_X^2+n_Y^2+n_Z^2)\pi^2\hbar^2}{2mL^2}$, $(n_X=1,2,3,\ldots;n_Y=1,2,3,\ldots;n_Z=1,2,3,\ldots)$.  Most of the energy levels given by this equation exhibit degeneracy: More than one quantum state has the same energy.

The hydrogen atom:  The Schrödinger equation for the hydrogen atom gives the same energy levels as the Bohr model: $E_n=-\frac1{(4\pi\epsilon_0)^2}\frac{m_\mathrm{r}e^4}{2n^2\hbar^2}=-\frac{13.60\,\mathrm{eV}}{n^2}$.  If the nucleus has charge $Ze$, there is an additional factor of $Z^2$ in the numerator.  The possible magnitudes $L$ of orbital angular momentum are given by equation: $L=\sqrt{l(l+1)}\hbar$, $(l=0,1,2,\ldots,n-1)$.  The possible values of the $z$-component of orbital angular momentum are given by equation: $L_z=m_l\hbar$, $(m_l=0,\pm1,\pm2,\ldots,\pm l)$.

The probability that an atomic electron is between $r$ and $r+dr$ from the nucleus is $P(r)\,dr$, given by equation: $P(r)\,dr=|\psi|^2\,dV=|\psi|^2\,4\pi r^2\,dr$.  Atomic distances are often measured in units of $a$, the smallest distance between the electron and the nucleus in the Bohr model: $a=\frac{\epsilon_0h^2}{\pi m_\mathrm{r}e^2}=\frac{4\pi\epsilon_0\hbar^2}{m_\mathrm{r}e^2}=5.29\times10^{-11}\mathrm{m}$.

The Zeeman effect:  The interaction energy of an electron (mass $m$) with magnetic quantum number $m_l$ in a magnetic field $\vec{B}$ along the $+z$-direction is given by equation: $U=-\mu_zB=m_l\frac{e\hbar}{2m}B=m_lm_{\mathrm{B}}B$ $(m_l=0,\pm1,\pm2,\ldots,\pm l)$, where $m_{\mathrm{B}}=\frac{e\hbar}{2m}$ is called the Bohr magneton.

Electron spin:  An electron has an intrinsic spin angular momentum of magnitude $S$, given by equation $S=\sqrt{\frac12(\frac12+1)}\hbar=\sqrt{\frac34}\hbar$.  The possible values of the $z$-component of the spin angular momentum are $S_x=m_s\hbar$ $(m_s=\pm\frac12)$.

An orbiting electron experience an interaction between its spin and the effective magnetic field produced by the relative motion of electron and nucleus.  This spin-orbit coupling, along with relativistic effects, splits the energy levels according to their total angular momentum quantum number $j$: $E_{n,j}=-\frac{13.60\,\mathrm{eV}}{n^2}[1+\frac{n^2}{\alpha^2}(\frac{n}{j+\frac12}-\frac34)]$.

Many-electron atoms:  In a hydrogen atom, the quantum numbers $n$, $l$, $m_l$, and $m_s$ of the electron have certain allowed values given by equation: $n\geq1$, $0\leq l\leq n-1$, $|m_l|\leq l$, $m_s=\pm\frac12$.  In a many-electron atom, the allowed quantum numbers for each electron are the same as in hydrogen, but the energy levels depend on both $n$ and $l$ because of screening, the partial cancellation of the field of the nucleus by inner electrons.  If the effective (screened) charge attracting an electron is $Z_{\mathrm{eff}}e$, the energies of the levels are given approximately by equation: $E_n=-\frac{Z_{\mathrm{eff}}^2}{n^2}(13.6\,\mathrm{eV})$.

X-ray spectra:  Moseley’s law states that the frequency of a $K_{\alpha}$ x ray from a target with atomic number $Z$ is given by equation $f=(2.48\times10^{15}\,\mathrm{Hz})(Z-1)^2$.  Characteristic x-ray spectra result from transition to a hole in an inner energy level of an atom.

Quantum entanglement:  The wave function of two identical particles can be such that neither particle is itself in a definite state.  For example, the wave function could be a combination of one term with particle $1$ in state $A$ and particle $2$ in state $B$ and one term with particle $1$ in state $B$ and particle $2$ in state $A$.  The two particles are said to be entangled, since measuring the state of one particle automatically determines the result of subsequent measurements of the other particle.