Energy level
From Wikipedia, the free encyclopedia
- This article is about orbital (electron) energy levels. For compounds' energy levels, see chemical potential.
A quantum mechanical system or particle that is bound, confined spacially, can only take on certain discrete values of energy, as opposed to classical particles, which can have any energy. These are called energy levels. The term is most commonly used for the energy levels (electron configuration) of electrons in atoms or molecules, which are bound by the electric field of the nucleus. In other words, their energy spectrum can be quantized (see continuous spectrum for the more general case).
If the potential energy is set to zero at infinity, the usual convention, then bound electron states have negative potential energy.
Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical state. They are then called degenerate energy levels.
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[edit] Explanation
Quantized energy levels result from the relation between a particle's energy and its wavelength. For a confined particle, the waves have the form of standing waves. Only stationary states with energies corresponding to integral numbers of wavelengths can exist; for other states the waves interfere destructively, resulting in zero probability density. Elementary examples that show how energy levels come about are the particle in a box and the quantum harmonic oscillator.
The following sections give an overview of the most important factors that determine the energy levels of atoms and molecules.
[edit] Atoms
[edit] Intrinsic energy levels
[edit] Orbital state energy level
Assume an electron in a given atomic orbital. The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by :
,
where 
is the Rydberg constant (typically between 1 eV and 103 eV), Z is the charge of the atom's nucleus, 
is the principal quantum number, e is the charge of the electron, h is Planck's constant, and c is the speed of light.
The Rydberg levels depend only on the principal quantum number .
[edit] Fine structure splitting
Fine structure arises from relativistic kinetic energy corrections, spin-orbit coupling (an electrodynamic interaction between the electron's spin and motion and the nucleus's electric field) and the Darwin term (contact term interaction of s-shell electrons inside the nucleus). Typical magnitude 10 − 3 eV.
[edit] Hyperfine structure
Spin-nuclear-spin coupling (see hyperfine structure). Typical magnitude 10 − 4 eV.
[edit] Electrostatic interaction of an electron with other electrons
If there is more than one electron around the atom, electron-electron-interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low.
[edit] Energy levels due to external fields
[edit] Zeeman effect
The interaction energy is: U = − μB with μ = qL / 2m
[edit] Zeeman effect taking spin into account
This takes both the magnetic dipole moment due to the orbital angular momentum and the magnetic momentum arising from the electron spin into account.
Due to relativistic effects (Dirac equation), the magnetic moment arising from the electron spin is μ = − μBgs with g the gyro-magnetic factor (about 2). μ = μl + gμs The interaction energy therefore gets UB = − μB = μBB(ml + gms).
[edit] Stark effect
Interaction with the external electric field causes:see Stark effect
[edit] Molecules
Roughly speaking, a molecular energy state, i.e. an eigenstate of the molecular Hamiltonian, is the sum of an electronic, vibrational, rotational, nuclear and translational component, such that:
where Eelectronic is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface) at the equilibrium geometry of the molecule.
The molecular energy levels are labelled by the molecular term symbols.
The specific energies of these components vary with the specific energy state and the substance.
In molecular physics and quantum chemistry, an energy level is a quantized energy of a bound quantum mechanical state.
[edit] Crystalline Materials
Crystalline materials are often characterized by a number of important energy levels. The most important ones are the top of the valence band, the bottom of the conduction band, the Fermi energy, the vacuum level, and the energy levels of any defect states in the crystals.
