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.

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