The simplest physical model of electron behavior in an atom is an electron in hydrogen. For a particle to remain in orbit, it must be bound to its center of rotation by some radial electric potential. In this system, electrons orbiting an atomic nucleus are bound to the nucleus via the
Coulomb potential, given by V(r) = \frac{e^2}{4 \pi \epsilon_0} \frac{1}{r}. Classically, the energy of the electron orbiting a nucleus would be given as the sum of the kinetic and potential energies. The Bohr model of a
hydrogen-like atom is a classical model of uniform circular motion. Its Hamiltonian is thus written in this way, as H_{Bohr} = \frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0}\frac{1}{r}. The first term is the kinetic energy of the electron (classically given as \frac{p^2}{2m}, where in quantum mechanics we have replaced classical momentum p with the momentum operator p = -i\hbar\nabla. The second term accounts for the Coulomb potential. The Bohr model energies, which are
eigenvalues of the Bohr Hamiltonian, are of order \alpha^2 mc^2, where \alpha is the unitless
fine-structure constant, defined as \alpha \equiv \frac{e^2}{4\pi\epsilon_0 \hbar c} \approx \frac{1}{137.036}. (Since \alpha is much less than 1, energy corrections with more factors of \alpha are significantly smaller order shifts). However, some revisions must be made to the simplified Bohr model of an electron in the hydrogen atom to account for quantum mechanical effects. These revisions to the electron's motion in a hydrogen atom are some of the most ubiquitous examples of quantum mechanical orbital motion. Ordered by greatest to smallest order of correction to the Bohr energies, the revisions are: • Motion of the nucleus (of order \alpha^4 mc^2) •
Fine structure (of order \alpha^4 mc^2) or the
Zeeman effect in the presence of a large magnetic field • Relativistic correction • Spin-orbit coupling •
Lamb shift (of order \alpha^5 mc^2): This is associated with the quantization of the electric field •
Hyperfine splitting (of order \frac{m}{m_p}\alpha^4 mc^2) For each revision, the Hamiltonian is first rewritten, and then the shifted energy levels are calculated using
perturbation theory.
Motion of the nucleus The nucleus is not really perfectly stationary in space; the Coulomb potential attracts it to the electron with equal and opposite force as it exerts on the electron. However, the nucleus is far more massive than the orbiting electron, so its acceleration towards the electron is very small relative to the electron's acceleration towards it, allowing it to be modeled as a also This is accounted for by replacing the mass (m) in the Bohr Hamiltonian with the reduced mass (\mu) of the system.
Fine structure • Relativistic correction: The first term in the Hamiltonian represents the kinetic energy of the electron in the atom. However, it comes from the classical expression for kinetic energy T = \frac{p^2}{2m}. However despite the fact that the electron is moving at relativistic speeds. The relativistic kinetic energy is given as the difference between the electron's total kinetic energy and its rest energy, T = \frac{mc^2}{\sqrt{1-(v/c)^2}} - mc^2. Expressing T in terms of the relativistic momentum of the electron and Taylor expanding in powers of the small number \frac{p}{mc}, yields a new expression for the kinetic energy which reduces to the classical term to first order: T = \frac{p^2}{2m} - \frac{p^4}{8m^3c^2} + .... Giving the lowest-order correction to the Hamiltonian as H_r ^\prime = - \frac{p^4}{8m^3c^2}. In non-denerate perturbation theory, the first order correction to energy levels is given by the expectation value of H ^\prime in the unperturbed state. E_r^1 = \langle \psi |H_r^\prime|\psi \rangle, and for the unperturbed states the Schrödinger equation reads p^2\psi = 2m(E-v)\psi. Putting these together, the correction to energy is E_r^1 = - \frac{1}{2mc^2}[E^2 - 2E \langle V\rangle + \langle V^2\rangle]. Substituting in the Coulomb potential and simplifying, we get E_r^1 = - \frac{(E_n)^2}{2mc^2} \lbrack\frac{4n}{l + 1/2} -3\rbrack. • Spin-orbit coupling: Each spin-1/2 electron behaves like a magnetic moment; the presence of a magnetic field exerts a torque which tends to align its magnetic moment \mu with the direction of the field. From the reference frame of the electron, the proton is circling around it; this orbiting positive charge creates a magnetic field
B in its frame. Relativistic calculations give the magnetic moment of the electron as \vec\mu_e = -\frac{e}{m}\vec S, where
S is the spin of the electron. The Hamiltonian for a magnetic moment is given as H = -\vec\mu \cdot \vec BThe magnetic field from the proton can be derived from the
Biot-Savart law, picturing the proton as a continuous current loop from the perspective of the electron: \vec B = \frac{1}{4 \pi \epsilon_0}\frac{e}{mc^2r^3}\vec L. With an extra factor of 1/2 to account for
Thomas precession, which accounts for the fact that the frame of the electron is non-inertial, the Hamiltonian is H_{so}^\prime = (\frac{e^2}{8\pi \epsilon_0})\frac{1}{m^2 c^2 r^3}\vec S \cdot \vec L. After some calculation of the eigenvalues of \vec S\cdot \vec L, the energy levels are found to beE_{so}^1 = \frac{(E_n)^2}{mc^2}\frac{n[j(j+1)-l(l+1)-3/4]}{l(l+1/2)(l+1)}. After accounting for all fine structure, the energy levels of the hydrogen-like atom are labeled as: E_{so}^1 = \frac{13.6\text{eV}}{n^2}[1 + \frac{\alpha^2}{n^2}(\frac{n}{j+1/2}- \frac{3}{4} )\rbrack
The Zeeman effect When an atom is placed in a uniform external magnetic field
B, the energy levels are shifted. This phenomenon shifts the Hamiltonian with the factor H_z^\prime = \frac{e}{2m}(\vec L+2 \vec S) \cdot \vec B_{ext}, where
L is the electron's angular momentum and
S is its spin. In the presence of a weak magnetic field, the fine structure dominates and the Zeeman Hamiltonian term is treated as the perturbation to the unperturbed Hamiltonian, which is a sum of the Bohr and fine structure Hamiltonians. The Zeeman corrections to the energy are found to be E_z^1 = \mu_B g_J B_{ext} m_j, where \mu_B\equiv \frac{e\hbar}{2m} = 5.788\times 10^-5 eV/T, is the
Bohr magneton. In a strong magnetic field, the Zeeman effect dominates and the unperturbed Hamiltonian is taken to be H_{Bohr}+ H_Z^\prime, with the correction H_{fs}^\prime. The corrected energy levels are labeled as: E_^1 = \frac{13.6\text{eV}}{n^3}\alpha^2\lbrack \frac{3}{4n} - (\frac{l(l+1)- m_l m_s}{l(l + 1/2)(l+ 1)})\rbrack.
Hyperfine splitting The proton also constitutes a weak
magnetic dipole, and
hyperfine splitting describes the effect is due to the interaction between the magnetic dipole moments of the electron and the proton. This effect gives rise to energy level shifts E_\text{hf}^1 = \frac{\mu_0 g_p e^2}{3 \pi m_p m_e a^3}\langle \vec S_p \cdot \vec S_e\rangle. It is called
spin–spin coupling because it involves the dot product between the spins of the electron and the proton. == Applications ==